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m  MEMOIRIAM 
Irving  Str Ingham 


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THE    NEW 

ADVANCED    ARITHMETIC 


BY 

JOHN   W.   COOK 

PRESIDENT    ILLINOIS    STATE    NORMAL    UNIVERSITY 
ANT) 

MISS   N.    CROPSEY 

ASSISTANT    SUPERINTENDENT    CITY    SCHOOL* 
INDIANAPOLIS,    INDIANA 


SILVER,    BURDETT   &   COMPANY 

New  York  .  .  .  BOSTON  .  .  .  Chicago 
1902 


t^ 


V 


Copyright,  189S,  1896, 
Bv  Silver,  Burdett  and  Company. 


H.  M.  PLIMPION  i.  CO..  PRINTERS  4  BINDERS, 
NORWOOD,  MASS.,  USA. 


PREFACE. 


IT  has  seemed  to  the  authors  of  the  J^grmal  Course 
IN  Number  that  there  is  room  for  another  series  uf 
Arithmetics,  notwithstanding  the  fact  that  there  are  many 
admirable  books  on  the  subject  already  in  the  field. 

The  Elementary  Arithmetic  is  the  result  of  the  ex 
perience  of  a  supervisor  of  primary  schools  in  a  leading 
American  city.  Finding  it  quite  impossible  to  secure 
satisfactory  results  by  the  use  of  such  elementary  arith- 
metics as  were  available,  she  began  the  experiment  of 
supplying  supplementary  material.  An  effort  was  made 
to  prepare  problems  that  should  be  in  the  highest  degree 
practical,  that  shonld  develop  the  subject  systematically, 
and  that  should  appeal  constantly  to  the  child's  ability 
to  think.  Believing  that  abundant  practice  is  a  prime 
necessity  to  the  acquisition  of  skill,  the  number  of  prob- 
lems was  made  unusually  large.  The  accumulations  of 
several  years  have  been  carefully  re-examined,  re-arranged, 
and  supplemented,  and  are  now  presented  to  the  public 
for  its  candid  consideration.  Not  the  least  valuable  fea- 
ture of  this  book  is  the  careful  gradation  of  the  examples, 
securing  thereby  a  natural  and  logical  development  of 
number  work.  No  space  is  occupied  with  the  presenta- 
tion of  theory,  —  that  side  of  the  subject  being  left  to 
the  succeeding  book.  The  first  thoughts  are  what  and 
Jww,  —  these    so    presented    that    the    processes    shall   be 

800574 


It  prefa  ce. 

easily  comprehended  and  mastered.  Subsequently,  the 
why  may  be  intelligently  considered  and  readily  under- 
stood. 

The  Advanckd  Arithmetic  is  the  outgrowth  of  a 
somewhat  similar  experience  in  the  class-room  of  a  teach- 
ers' training-school.  For  many  years  an  opportunity  was 
afforded  to  study  the  effects  upon  large  numbers  of  pu- 
pils of  the  current  methods  of  instruction  in  arithmetic. 
The  result  of  such  observation  was  the  conviction  that 
the  rational  side  of  the  subject  is  seriously  neglected. 
An  effort  was  made  to  supplement  the  ordinary  text-book 
by  a  study  of  principles  and  by  explanations  of  pi'ocesses. 
The  accumulations  of  fifteen  years  have  been  edited  with 
all  of  the  discrimination  of  which  the  authors  were  capa- 
ble. Great  care  has  been  exercised  in  the  presentation  of 
principles  and  in  the  formulation  of  processes,  to  the  end 
that  the  learner  shall  have  every  facility  for  the  use  of 
his  reasoning  powers,  and  at  no  point  be  relieved  from 
the  proper  exercise  of  his  mental  activity  and  acumen. 
It  is  hoped  that  tlie  book  may  contribute  somewhat  to 
the  movement,  now  so  happily  going  oil,  that  looks 
toward  the  disestablishment  of  the  method  of  pure  au- 
thority, and  the  establishment  of  a  method  that  makes 
its  appeal  to  intelligence  and  reason. 

The  authors  desire  to  express  tlieir  appreciation  of  the 
excellent  suggestions  offered  by  many  friends;  but  espe- 
cial tlianks  are  due  Professor  David  Felmley,  of  the 
Illinois  State  Normal  School,  for  his  discriminating 
criticisms  and   valuable  assistance. 

THE    AUTHORS. 


SUGGESTIOIS^S. 


METHOD  is  determined  chiefly  by  aim.  The  answer 
whicli  the  teacher  makes  to  the  question,  "  Why 
should  boys  and  girls  study  arithmetic?"  will  guide  him 
in  the  details  of  instruction. 

Arithmetic  is  one  of  the  traditional  "three  R's."  Some 
knowledge  of  its  processes  is  necessary  to  any  degree  of 
intelligence.  Its  highly  practical  character  is  conceded 
by  every  one. 

The  arithmetical  operations  employed  in  ordinary  busi- 
ness affairs  are  simple,  but  they  must  be  performed  with 
absolute  accuracy  and  with  great  rapidity.  They  are 
based,  primarily,  upon  the  memory.  The  necessity  for 
perfect  familiarity  with  the  fundamental  facts  of  number 
becomes  apparent.  Neither  accuracy  nor  rapidity  is  poS' 
sible  without  a  thorough  mastery  of  the  primary  work. 
This  mastery  is  acquired  through  constant  repetition  of 
the  old  in  connection  with  the  acquisition  of  the  new. 
One  of  the  teacher's  maxims,  constantly,  must  be  "Ee- 
viewf    Review!!    REVIEW!!! 

But  arithmetic  has  another  and  a  higher  function.  It 
must  cultivate  t^at  quick  intelligence  which  is  able  to 
analyze  given  conditions  and  determine  what  should  b& 
done  in  the  particular  case. 

A  true  problem  in  arithmetic  is  a  statement  or  seriea 
of  statements  in  which  something  is  told  and  somethiag 


vi  aUGGESTIONS. 

is  asked  ;  the  answer  to  what  is  asked  being  implied  iv 
what  is  told.  The  chief  activity  of  the  child,  in  the  solu- 
tion of  true  i)robleius,  is,  first,  tlie  analyzing  of  given  con 
ditions,  and  second,  the  performing  of  certain  operations. 
The  former  of  these  activities  should  never  become  me- 
chanical; the  latter  should  pass  into  the  mechanical  stage 
as  early  as  possible. 

The  study  of  advanced  arithmetic  differs  in  certain 
essential  particulars  from  all  other  studies  in  the  common- 
school  course.  The  thought  lacks  the  continuity  of  his- 
tory or  geography.  The  work  consists  of  a  series  of  efforts 
that  are  more  or  less  distinct.  Each  problem  stands  alone 
in  its  statement;  but  the  operations  involved  in  it  fall 
under  certain  general  types.  Arithmetic,  consequently, 
requires  a  constant  dealing  with  both  particular  and  gen- 
eral notions.  These  general  processes  may  be  discovered 
in  illustrative  problems,  and  the  possible  number  of  them 
be  absolutely  exhausted.  When  this  has  been  done,  tlie 
"case"  arrangement  of  problems  should  cease,  and  the 
pupils  should  be  permitted  to  perform  that  analytic  ac- 
tivity that  discovers  conditions  in  a  problem,  and  that 
synthetic  activity  that  unites  it  to  its  proper  class.  The 
problems  in  this  book  have  been  prepared  with  this 
thought   in    mind. 

Another  phase  ot  arithmetic  work  should  be  clearly 
appreciated.  It  surpasses  all  other  studies  in  the  low;er 
grades  in  the  number  of  its  generalizations  and  the  ease 
with  which  they  are  made.  Tlie  generally  accepted  prin- 
ciples tnat  instruction  should  proceed  from  the  concrete 
to  the  abstract,  and  from  the  individual  to  the  general, 
have  constant  application.  But  the  abstract  should  be 
the   result  of  a   conscious    abstraction,    and    the    gwieraL 


SUGGESTIONS.  vu 

should  be  a  conscious  generalization.  As  illustrating  this 
tliought,  and  showing  how  the  general  is  reached  from 
the  individual,  particular  attention  is  called  to  the  method 
of  deriving  rules  from  processes.  The  formulation  and 
statement  of  a  rule  is  a  generalization. 

It  is  a  fundamental  principle  of  true  teaching,  that 
whatever  is  done  by  the  pupil  shall  be  accomplished, 
through  his  conscious,  personal  effort,  and  that  whatever 
his  acquisition,  it  shall  be  consciously  his  own,  —  not  his 
in  the  memory  -alone,  which  is  the  same  as  his  in  the 
book,  but  his  vitally  and  substantially,  as  the  blood  in 
his  veins  or  the  innate  ideas  of  right  and  wrong.  Such 
knowledge  is  of  a  rooted  and  growing  order  that  gives 
satisfaction  and  power  to  its  possessor. 

To  suggest  and  stimulate  such  teaching,  and  to  secure 
such  growth,  many  questions  are  asked  and  many  direc- 
tions are  given,  in  this  book,  which  are  designed  to  throw 
the  pupil  upon  his  own  resources.  The  questions  cannot 
be  answered  by  any  statements  found  on  the  pages,  —  no 
need  to  tax  the  memory  for  words  and  phrases ;  yet 
all  the  questions  and  directions  are  simple  and  easily  to 
be  answered  by  the  pupil,  if  he  has  thought  his  way 
clearly. 

Too  great  emphasis  cannot  be  given  to  the  statement 
already  made,  that  the  work  of  analyzing  should  never 
become  mechanical.  While  there  is  value  in  concise  and 
definite  formulas,  there  is  infinitely  greater  value  in  free- 
dom. The  mind  should  be  free  to  discover  conditions, 
relations,  and  sequences;  it  should  be  as  free  in  stating 
conclusions  and  results ;  but  this  freedom  cannot  exist  if 
the  reasoning  is  compelled,  per  force,  to  follow  a  memo- 
rized formula.     The  forms   of  analyses  given  in  the  fol- 


viii  SUGGESTIONS. 

lowing  pages  are  presented  as  models  to  be  studied  and 
mastered,  but  not  to  be  memorized.  If  the  pupil,  in  his 
own  way  and  in  different  words,  shall  clearly  present  the 
steps  of  reasoning  and  draw  the  correct  conclusion,  his 
work  should  be  approved. 

An  additional  suggestion  must  suffice.  The  teacher's 
"knowledge  of  the  subject  should  be  organic.  Arithmetic 
should  be  recognized  as  a  science  that  is  deduced  from 
the  idea  of  addition.  When  the  so-called  fundamental 
processes  have  been  mastered,  little  remains  but  repeti- 
tion. The  fraction  differs  from  the  integer  in  that  it  in- 
troduces a  double  unity.  The  decimal  fraction  differs 
from  the  common  fraction  in  the  method  of  expressing 
its  denominator  Percentage  is  "a  case"  in  decimal  frac- 
tions. Compound  numbers  differ  from  simple  numbers 
because  of  the  introduction  of  variable  scales,  etc.,  etc. 
As  the  power  to  generalize  relieves  the  mind  from  the 
overwhelming  burden  of  a  countless  multitude  of  indi- 
viduals, so  each  new  step  in  advance  is  easily  held  if 
correllated  with  the  fundamental  ideas. 

J.  W.  C. 


CONTENTS. 


PAKT   I. 


Section  I. 

Definitions     .     , 

Notation    .     .     , 
Numeration  . 
Roman  Notation 
Reduction     .     . 


Section  II. 
Addition   . 


Section  III. 
Subtraction 


Section  IV, 

Multiplication  .... 
Multiplication  by  Factors 
Surface  Measure    .     .     . 

Section  V. 

Division 

Long  Division  .... 
Division  by  Aliquol;  Parts 
Division  by  Factors  .  . 
Law  of  Signs  .... 
Properties  of  Numbers  . 
Tests  of  Divisibility  ,  . 
Factoring  ..... 
Cancellation 


11 


22 


45 
49 
54 
55 
59 
65 
67 
72 
73 


FAoa 


Section  VI. 

Fractions  .               .    .     .    . 

76 

Reduction 

78 

Addition  of  Fractions     .     . 

83 

Least  Common  Multiple 

83 

Subtraction  of  Fractions     . 

90 

Multiplication  of  Fractions 

96 

Division  of  Fractions      .     . 

103 

Complex  Fractions     .     .     . 

108 

Decimal  Fractions      .    . 

125 

Numeration  of  Fractions 

126 

Notation  of  Fractions 

127 

Reduction  of  Decimals  • 

129 

Addition  of  Decimals 

132 

Subtraction  of  Decimals 

133 

Multiplication  of  Decimals 

135 

Division  of  Decimals  .     . 

138 

Measurement  of  the  Circle 

143 

Federal  Money  .... 

.     147 

Bills  and  Statements  .     . 

148 

Denominate  Numbers     . 

151 

Measures  of  Length   .     . 

.     151 

Surface  Measure    .     .     . 

.     156 

Surveyor's  Measure    . 

.     159 

United  States  Surveys    . 

.     160 

Measures  of  Volume  .     . 

.     162 

Wood  Measure      .     .     . 

.     163 

Lumber  Measure    .     .     . 

.     165 

Measures  of  Capacity     . 

.     167 

Weight     .     . 

.     168 

CONTENTS. 


English  Money 

French  Money  .     . 

German  Money 

Circular  Measure  . 

Longitude  aud  Time 

The  International  Date  Lite 


Page 
172 
172 
173 
173 
174 
'80 

Calendar 182 

Miscellaneous  Tables      .     .     183 

PART  IL 
Section  VII. 
Percentage 203 

Section  VIII. 

Applications  of  Percentage  224 

Profit  and  Loss      ....  224 

Commission 231 

Commercial  Discount      .     .  234 
Stocks,  Bonds,  and  Broker- 
age      236 

Taxes 242 

United  States  Revenue  .     .  245 

Insurance 248 

Property  Insurance    .     .     .  249 

Life  Insurance 250 

Interest 255 

Accurate  Interest  ....  266 

Partial  Payments  ....  273 

Compound  Interest     .     .     .  274 
General  Problems  in  Simple 

Interest 278 

Present    Worth    and    True 

Discount 2d-2 

Bank  Discount 285 

Exchange 289 

Foreign  Exchange      .     .     .  293 

Equation  of  Pa\'ments    .     .  295 


Section'  IX. 

I^atio 306 

Proportion 307 

Compound  Ratio    .     .     .     .  311 

Partnership 314 

Section  X. 
Involution  .  .  ...  317 
Evolution  ...  .  .  318 
The  Right  Triangle  .  .  .  324 
Cube  Root  .....  328 
Cube  Roof  of  Decimal  Frac- 
tions        331 

Cube  Root  of  Common  Frac- 
tions       332 

The  Cone 341 

Prisms  and  Pyramids  .  .  S42 
General  Reviews  ....  344 
Algebraic  Questions  .  .  .  356 
The  Literal  Notation  .  .  356 
Evaluation  of  Algebraic  [ex- 
pressions     358 

Axioms 361 

Equations  containing  Frac- 
tions       364 

Positive  and  Negative  Quan- 
tities       369 

Law  of  Signs  in  Division    .     375 

Appendix. 

Greatest  Common  Divisor  .  379 

Least  Common  Multiple     .  384 

Mariner's  Measure  .     .  386 

Average  of  Accounts      .     .  386 

Origin  of  Units       ....  387 
Metric  System  of  Weights 

and  Mea.«iirps      ....  388 


THE    NORMAL    COURSE    IN    NUMBER 


THE  NEW 
ADVANCED   ARITHMETIC. 


^att  I. 

SECTION    L 

DEFINITIONS. 

1.  Measuring  is  the  process  of  finding  how  many  times  a 
quantity  contains  a  part  of  itself  which  is  taken  as  a  stand- 
ard.    lUustrate. 

2.  That  portion  of  a  measured  quantity  which  is  used  as 
a  standard  is  called  a  Unit. 

When  we  count  a  basket  of  eggs,  we  measure  the  quantity  of 
eggs  by  using  one  of  them  as  a  standard.  Such  a  unit  is  a  natural 
unit.  When  we  measure  a  quantity  of  cloth,  we  use  a  portion  of 
itself,  called  a  yard,  as  a  standard.  Such  a  unit  is  an  artificial 
unit. 

3.  From  the  repetitions  of  the  unit  in  counting  or  measur- 
ing, the  successive  numbers,  one,  two,  three,  etc,  arise ;  thus, 

Number  is  that  which  answers  the  question  "  How  many  ?  " 

4.  Arithmetic  is  the  science  which  treats  of  number  and 
the  methods  of  employing  it  in  computation. 

5.  To  simplify  counting  or  measuring,  units  are  gathered 
into  equal  groups,  each  of  which  forms  a  new  unit.  Thus,  in 
measuring  a  basket  of  eggs,  they  are  grouped  into  dozens. 
The  quantity  may  be  expressed  as  a  number  of  single  eggs 
or  as  a  number  of  dozens. 

1 


2  NEW  ADVANCED  ARITHMETIC. 

A  Decimal  System  of  numbers  is  a  system  in  which  ones  are 
grouped  into  tens ;  tens  into  tens  of  tens,  or  hundreds ;  hundreds 
into  tens  of  hundreds,  or  thousands ;  thousands  into  ten-thousands, 
etc. 

Ones  are  units  of  the  first  rank,  or  order ;  tens  are  units  of  the 
second  order ;  hundreds,  of  the  third  order,  etc. 

Illustrate  these  groupings  with  bundles  of  splints. 

The  scale  in  any  system  of  numbers  is  the  numbei  of  units  in 
each  order  required  to  form  one  of  the  next  higher. 

NOTATION. 

6.  The  art  of  expressing  numbers  by  means  of  characters 
is  Notation. 

7.  A  system  of  Notation  which  will  express  all  nuipbers 
must  include  a  set  of  characters  to  represent  numbers  and 
the  laws  for  using  them. 

8.  The  Arabic  System  of  Notation  employs  ten  charac- 
ters called  figures.  They  are  1  (one),  2  (two),  3  (three), 
4  (four),  5  (five),  6  (six),  7  (seven).  8  (eight),  9  (nme), 
0  (cipher). 

9.  Any  given  figure  always  expresses  th^  same  number 
of  units,  but  the  order  or  kind  of  units  is  expressed  by  the 
place  in  w^hich  the  figure  is  written. 

Ones  stand  in  the  first  place,  tens  in  the  next  place  to  the  left, 
hundreds  in  the  third  place,  etc.  ;  thus, 

A  figure  standing  in  any  place  expresses  units  ten  times  as  large 
as  if  standing  one  place  to  the  right 

The  cipher  is  used  to  fill  vacant  orders. 

NUMERATION. 

10.  The  art  of  reading  numbers  expressed  by  figures  ki 
Numeration. 

11.  Three  orders  form  a  period.  The  name  of  the  lowest 
order  in  each  period  is  ones ;  of  the  next  higher  is  tens ;  of 
the  third  is  hundreds. 


NUMERATWX.  8 

The  names  of  the  first  twelve  periods  in  their  order  are  as 
follows : 

1.  Units.  2.  Thousands.  3.  Millions.  4.  Billions.  5.  Tril- 
lions. 6.  Quadrillions.  7.  Quintillions.  8.  Sextillions.  9.  Sep- 
tillions.     10.    Octillions.     11.  Nonillions.     12.    Decillions. 

Note.  —  The  meaniug  of-the  prefix  iu  the  word  biUlon  is  two;  iu  tlie 
word  trillion  is  three  ;  in  quadrillion  is  four,  aud  so  on.  Observe  that  the 
number  of  auy  period  above  millions  is  two  more  than  the  meaning  of 
the  prefix  in  the  uame  of  that  period. 

12.    Arraugement  of  orders  aud  periods. 

Trillions.        Billions.        Millions.     Thousands.       Units. 


865,406,38    2,104,579 

Note.  —  Learn  the  names  of  the  periods  in  their  order  from  left  to 
right. 

13.  To  read  a  number,  group  the  figures  into  periods,  be- 
ginuing  at  the  right,  and  separating  tlie  periods  by  commas. 
Beginning  at  the  left,  read  the  number  in  each  period  as  if  it 
stood  alone ;  then  add  the  name  of  the  period. 

Note.  — The  English  system  of  Numeration  is  in  use  in  England  and 
upon  tlie  continent  of  Europe,  except  in  France.  It  employs  six  orders 
for  a  period.  In  studying  this  system  the  meaning  of  the  names  of  the 
periods  is  made  plain.  A  million  is  a  thousand  thousand.  A  billion  is  the 
square  of  a  million  ;  a  trillion,  the  third  power  of  a  million;  a  quadrillion, 
the  fourth  power  of  a  million,  etc. 

14.  Eead  the  following  numbers : 


1. 

2345. 

6. 

250849. 

11. 

683471. 

2. 

4638. 

7. 

381307. 

12. 

829406. 

3. 

7912. 

8. 

408391. 

13. 

200619. 

4. 

3105. 

9. 

716004. 

14. 

100054. 

5. 

26853. 

10. 

5Q0836. 

15. 

973070. 

16. 

253087G. 

23. 

17. 

3890432. 

24. 

18. 

470G3502. 

25. 

19. 

50780439. 

26. 

20. 

480983048. 

27. 

21. 

379068452016. 

28. 

22. 

690750142953. 

29. 

A^£;H''  .4Z)r.4.VC£i)  ARITHMETIC. 

85968345620961. 
6390086133859016. 
7000492587291563295. 
8230075913748426950. 
27.  2005300861943186627. 

10030006729062127390037. 
300750916400853269057040. 

]S;oTE.  —  Freciueiit  dictatiou  exercises  should  be  given  with  successively 
larger  numbers  until  pupils  have  acquired  proficiency  in  writing  numbers. 

15.  Before  writing  the  following  number's,  tell  how  each 
will  appear  when  written. 

Illustration.  Three  thousand  eight  hundred  seven  is  ex- 
pressed by  writing  the  following :  three,  comma,  eight,  cipher, 
seven. 

1.  Forty  thousand  six. 

2.  Ninety-seven  thousand  five  hundred  twelve. 

3.  Three  hundred  sixty-nine  thousand  twenty- four. 

4.  Four  million  eight  thousand  two. 

5.  Fifty-six  million  nineteen  thousand  thirty-three. 

6  Eighty-one  million  five  hundred  thirteen  thousand  two 
hundred  fifty-one. 

7o    Three  hundred  million  ninety  thousand  four. 
8     Five  billion  six  million  seven  thousand  eight. 

9.  Seventy -two  billion  six  hundred  thirty-five  thousand 
two  hundred  fifty-one. 

10.  One  hundred  three  billion  two  nuUion  seventeen 
thousand  one  hundred  four. 

11  Two  trillion  three  billion  four  million  five  thousand  six. 

12  Nmety-one  trillion  two  hundred  seven  billion  sixty- 
nine  million  four  thousand  three 

13  Eighty-six  trillion  one  million  twenty-three. 


NUMERA  TION.  5 

14.  Two  hundred  sixteen  trillion  five  hundred  thousand. 

15.  Nineteen  trillion  four. 

XuTE.  1.  Teachers  should  supply  dictation  exercises  in  writing  nam- 
bers  until  a  good  degree  of  proficiency  is  acquired. 

2.  Observe  which  of  the  number  names  are  eompound  words. 
3    Note  that  the  word  "  aud  "  is  not  used  in  these  exercises. 

16.     THE   ROMAN    NOTATION. 

This  method  expresses  number  by  the  use  of  certain  print 
letters.     They  are  I,  V,  X,  L,  C,  D,  M. 

1  =  1,  V  =  5,  X=10,  L  =  50,  C  =  100,  D  =  500, 
M  =  1000.     Their  use  is  determined  by  the  following 

PRINCIPLES. 

1.  Repeating  a  letter  repeats  its  value.    II  =  2,  XX  =  20. 

2,  When  a  letter  is  placed  after  one  of  greater  value,  the 
tTso  express  a  number  equal  to  the  sum  of  their  values. 
XV  =:  15,  CI=  101. 

3  When  a  letter  is  placed  before  one  of  greater  value, 
the  two  express  a  number  equal  to  the  difference  of  their 
values.  IX  =  0,  XC  =  90.  (Limited  to  IV,  IX,  XL, 
and  XC.) 

4.  When  a  letter  is  placed  between  two,  each  of  greater 
value,  its  value  is  taken  from  the  sum  of  their  values. 
XIX  r=  19,  XIV  =  14. 

5.  Placing  a  dash  over  a  letter  multiplies  its  value  by  a 
thousand      X  =  10,000,  M  =  1,000,000. 

EXERCISES 
1    Express  by  the  Roman  characters  all  numbers  from  one 
to  one  hundred 

2,  125.  5    419c  8. 

3,  263c  6.    599o  9. 

4,  379  7.    648.  10. 

Note     Use  dictation  exercises  freely. 
2A 


752 

11. 

1776. 

1066. 

12. 

1799. 

1492. 

13. 

1896. 

6  NEW  ADVANCED  ARITHMETIC. 

17.     REDUCTION. 

1.  Reduction  is  the  process  of  changing  the  unit  of  a 
number  without  changing  its  value. 

2.  Express  6  pints  in  quarts ;  8  quarts  in  gallons ;  6  feet 
in  yards ;  32  ounces  in  pounds ;  20  mills  in  cents ;  300  cents 
in  dollars. 

3.  Have  tliese  numbers  been  changed  to  a  larger  or  to  a 
smaller  unit? 

The  process  of  reducing  a  number  to  larger  units  is  called 
Reduction  Ascending. 

4.  How  are  pints  reduced  to  quarts?  quarts  to  gallons? 
feet  to  yards?  ounces  to  pounds?  mills  to  cents?  ones  to 
tens?  tens  to  hundreds?  tens  to  thousands?  cents  to  dol- 
lars?    Give  many  similar  examples. 

DIRECTION. 

5.  To  reduce  a  ntitnber  to  a  tiu/tnber  of  larger  miits, 
divide  it  by  the  number  of  the  given  units  which  -makes 
one  of  the  larger  units. 

6.  Express  3  quarts  in  pints;  3  gallons  in  quarts;  4  yards 
in  feet;  3  dimes  in  cents;  5  tens  in  ones;  6  hundreds  in 
tens;    7  thousands  in  hundreds,  in  tens. 

7.  Have  these  numbers  been  changed  to  a  larger  or  to  a 
smaller  unit? 

The  process  of  reducing  a  number  to  smaller  units  is  called 
Reduction  Descending. 

8.  How  are  quarts  reduced  to  pints?  gallons  to  quarts? 
yards  to  feet?  dimes  to  cents?  tens  to  ones?  hundreds  to 
tens?    thousands  to  luuidreds?    to  tens? 

DIRECTION. 

9.  To  reduce  a  number  to  a  number  of  smaller  unitSf 
multiithi  it  bi/  the  number  of  tlie  smallet  units  to  which 
one  of  the  larger  units  is  equal. 


REDUCTION.  7 

Note.  —  A  knowledge  of  the  fundamental  nature  of  the  decimal 
STStem  is  of  the  utmost  importance  in  aritlniietical  operations.  This  is 
obtained  through  ]iractice  in  the  two  forms  of  Keduction.  Multiply  ex- 
amjiles  like  the  following 

18.  1.  In  50,000  there  are  how  many  tens?  hundreds? 
thousands  ?   ten-thousands  ? 

2.  In  17,000,000  there  are  how  many  thousands?  hun- 
dred-thousands?   tens?    hundreds?    ones?    ten-thousands? 

3.  In  38,000  mills  there  are  how  many  cents?  dimes? 
dollars  ? 

4.  In  65  dollars  there  are  how  man}'  dimes?  cents? 
mills? 

19.  1  is  what  part  of  10?  One  ten  is  what  part  of  100? 
One  hundred  is  what  part  of  1000?  In  1,111  each  unit  is 
what  part  of  the  unit  standing  in  the  first  place  at  its  left^ 

20.  It  is  customary  to  fix  the  place  of  ones  by  placing  a 
period  at  its  right,  thus :  1 .  When  the  period  is  thus  used 
it  is  called  the  decimal  point.  From  what  has  been  observed 
what  kind  of  unit  will  the  right-hand  figure  in  1.1  express? 
What  is  r)x\Q  tenth  of  one  tenth?  What,  then,  is  the  kind  of 
unit  expressed  by  the  right-hand  figure  in  1.11?  What  is 
one  tenth  of  one  hundredth?  What  kind  of  unit  is  ex- 
pressed by  the  right-hand  figure  in  1,111?  Similarly,  show 
what  kind  of  units  is  expressed  by  a  figure  in  the  fourth 
place  at  the  right  of  tile  decimal  point;  in  the  fifth  place; 
in  the  sixth  place. 

21.  Table  of  six  places  at  the  right  of  the  decimal  point. 


c     §     o     c     5     a 

H      W      H      H      W      S 

.236418 


t 

.6 

11. 

2. 

.8 

12. 

3. 

.2 

13. 

4. 

.24 

14. 

5. 

.37 

15. 

6. 

.04 

16. 

7. 

ol27 

17. 

8. 

.209 

18. 

9. 

.842 

19. 

10. 

.094 

20. 

31. 

16.07  (Read  : 

32. 

25.375 

i-LlALi 

33. 

239.004 

34. 

508.0089 

35. 

2851.3675 

36. 

3750.1049 

21. 

.00836 

22, 

.04826 

23. 

.39627 

24. 

.30861 

25. 

.000003 

26. 

.000072 

27. 

.000739 

28. 

004936 

29. 

.003972 

30. 

.386492 

8  NEW  ADVANCED  ARITHMETIC. 

22.  The  name  of  any  number  is  the  same  as  the  place 
in  which  its  right-hand  figure  stands.  Read  the  follov/ing 
numbers : 

.007 
.491 
.0005 
o0036 
.0683 
.2758 
.3085 
.4902 
.00006 . 
.00034 

16  and  7  hundredths  or  as  1607 
hundredths.) 

37.  6.00039 

38.  28643907502 

39.  10000.807501 

40.  .197527 

23.  Perfect  familiarity  with  the  names  and  numbers  of 
the  places  at  the  right  of  the  decimal  point  is  necessary  for 
accurate  and  rapid  writing.  The  name  of  a  number  is  de- 
termined by  the  place  of  its  right-hand  figure.  In  writing 
numbers  like  the  foregoing  correctly  the  first  time  two  things 
must  be  known :  the  number  of  figures  required  to  express 
the  number;  the  place  in  which  the  right-hand  figure  must 
stand  to  express  the  kind  of  units. 

24.  Write  the  following  numbers : 

1.  Twenty-three  hundredths. 

2.  Seven  tenths. 

3.  Sixty-nine  hundredths. 


NOTATION.  9 

4.  One  hundred  forty-eight  thousandths. 

5.  Two  hundred  eleven  thousandths. 

6.  Six  hundred  ninetj'-three  thousandths. 

7.  Nine  ten-thousandths. 

8.  Four  ten-thousandths. 

9.  Thirt3'-two  ten-thousandths. 

10.  One  hundred  eighty-three  ten-thousandths. 

11.  Nine  hundred  seven  ten-thousandths 

12.  One   thousand    five    hundred    seventy=six    ten    thou- 
sandths. 

13.  Nine  thousand  four  hundred  three  ten-thousandths. 

14.  Eight  hundred-tliousandths. 

15.  Five  hundred- thousandths. 

16.  Forty-five  hundred-thousandths. 

17.  Ninety-four  hundred-thousandths. 

18.  Three  hundred  fifty-two  hundred-thousandths. 

19.  Eight  hundred  ten  hundred-thousandths. 

20.  Four  thousand  seven  hundred  nineteen  hundi-ed- thou- 
sandths. 

21.  Seven  thousand  twenty-three  hundred-thousandths. 

22.  Two  thousand  sis  hundred-thousandths. 

23.  Thirty-four  thousand  six  hundred  twentj^-one  hundred- 
thousandths. 

24.  Fifty-nine  thousand  one  hundred  six  hundred-thou- 
sandths. 

25.  Nineteen  thousand  three  hundred-thousandths. 

26.  Fifty  thousand  one  hundred-thousandths. 

27.  25  millionths.  32.    5008  millionths. 

28.  8  millionths.  33.    37592  millionths. 

29.  63  millionths.  34.    8090G6  millionths. 

30.  478  millionths.  35.    26  and  68  hundredths. 

31.  2895  millionths.  36.    94  and  39  thousandths. 


10  NEW  ADVANCED  ARITHMETIC, 

37.  290  aud  463  hundred-thousandths. 

38.  40073  and  50093  millionths. 

39.  61  aud  13  millionths. 

40.  Two  thousand  five  and  three  thousand  forty-six  mil- 
lionths. 

25.  Numbers  like  exercises  1  to  30,  Art.  22,  are  called 
Decimal  Fractions.  Numbers  31  to  39  are  called  Mixed 
Decimals.  Name  the  Decimal  Fractions  in  Art.  24.  Name 
the  Mixed  Decimals  in  the  same  section. 

26.     REDUCTIONS. 

1.  In  .6  there  are  how  many  hundredths?  How  many 
thousandths?  Write  each  of  the  numbers.  How  many  ten- 
thousandths?  Write  the  number.  How  many  millionths? 
Write  the  number.  Read  3.o  as  hundredths,  as  thousandths, 
ten-thousandths,  hundred-thousandths,  millionths.  Write 
each  of  the  numbers.  In  .600000  there  are  how  many 
tenths?  ten-thousandths?  hundredths?  hundred-thou- 
sandths ?  millionths  ? 

2.  In  3.2  there  are  how  many  hundredths?  How  many 
ten-thousandths?  How  many  millionths?  Write  each  of 
the  numbers.  In  5.000000  how  many  hundred-thousandths? 
hundredths?  thousandths?  tenths?  Write  each  of  the 
numbers. 

3.  In  400000,  there  are  how  many  tens?  How  many 
hundreds?  How  many  ten-thousands?  How  many  thou- 
sands? 


ADDITION.  11 

SECTIOX    11. 

ADDITION. 

27.  Like  numbers  are  numbers  that  are  made  by  repeating 
the  same  unit. 

28.  The  Sum  of  two  or  more  like  numbers  is  a  number 
which  contains  all  of  their  units. 

29.  The  sign  +  Cpl^s)  between  two  numbers  shows  that 
their  sum  is  to  be  found. 

30.  At  first,  to  unite  two  numbers,  we  count  the  units  of 
the  second  upon  the  first,  thus :  In  uniting  four  blocks  and 
three  blocks,  we  think  (or  say)  four,  five,  six,  seven,  as  we 
place  the  three  blocks  one  at  a  time  with  the  four.  In  this 
way,  hy  counting  objects,  we  learn  and  commit  to  memory 
the  following  forty-five  sums: 

12  3         2         4  3         5         4         3 

1112  12  12         3 


6 

5 

4 

7 

G 

5 

4 

8 

7 

1 

2 

3 

1 

2 

3 

4 

1 

2 

— 

— 

- 

— 

— 

— 

■— 

6 

5 

9 

8 

7 

6 

5 

9 

8 

3 

4 

1 

o 

3 

4 

5 

2 

3 

— 

- 

— 

— 

— 

— 

~™ 

"" 

7 

6 

9 

8 

7 

6 

9 

8 

7 

4 

5 

3 

4 

5 

6 

4 

5 

6 

— 

— 

— 

— 

— 

— 

~ 

— 

— 

9 

8 

7 

9 

8 

9 

8 

9 

9 

5 

6 

7 

n 

7 

7 

8 

8 

9 

31.    This  series  of  forty-five  sums  of  the  nine  primary  num- 
bers taken  in  twos  is  called  the  Addition  Table. 


12  NEW  ADVANCED  ARITHMETIC. 

The  last  twenty  sums  in  the  addition  table  may  be  learned 
without  objects  by  uniting  with  the  first  number  enough  ot  the 
second  to  make  ten ;  thus,  8  +  5  =  8  +  2  +  3  =  10 +  3  =  13. 

32.  Addition  is  the  process  of  finding  the  sum  of  two  or 
more  like  numbers. 

Since  7  +  5  =  12,  17  +  5  =  22,  27  +  5  =  32,  etc.  Practice  sim- 
ilarly with  all  of  the  "  endings." 

33.  As  a  preparation  for  "  column  addition"  give  frequent 
exercises  in  adding  by  twos,  threes,  fours,  etc. ,  starting  with 
one,  two,  three,  etc.,  and  carrying  the  work  to  50. 

34.    EXERCISES. 

The  following  problems  are  for  seat,  board,  or  home  work, 
and  for  oral  practice  in  recitation.  Practice  upon  them  until 
they  can  be  performed  with  great  rapidity.  They  contain 
frequent  repetitions  of  the  work  in  the  addition  table.  Add 
from  the  bottom,  naming  results  only,  thus:  (first  problem) 
7,  13,  21,  24,  26,  etc.  To  test  the  accui-acy  of  the  w^ork  add 
downward. 

(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 
957  5  968649 
5678887985 
7565778  2  99 
685998  2  763 
274  3  676972 
3  897963786 
8436356839 
69669    5    7499 


(11) 

(12) 

(13) 

(14) 

(1.5) 

(16) 

(17) 

(18) 

89 

79 

38 

93 

37 

43 

93 

98 

29 

63 

34 

84 

28 

92 

89 

79 

93 

94 

34 

68 

22 

99 

83 

78 

26 

33 

68 

89 

21 

99 

83 

89 

97 

29 

93 

62 

88 

78 

29 

35 

96 

98 

82 

99 

83 

87 

39 

73 

ADDITION. 

m 

(20) 

(21) 

(22) 

324 

752 

945 

8685 

561 

433 

887 

4944 

872 

512 

654 

5636 

324 

311 

472 

3768 

607 

721 

541 

9483 

830 

869 

635 

7521 

962 

863 

796 

2754 

574 

127 

4856 

385 

7039 

(23) 

(24) 

(25) 

(20) 

3580 

2816 

77281 

497529 

6295 

6389 

94969 

678315 

3782 

3528 

8S799 

855274 

9061 

4893 

68698 

998865 

3648 

4678 

89769 

677345 

7296 

8253 

84858 

499586 

6802 

7594 

48979 

394759 

3690 

6031 

87857 

862979 

3285 

8539 

36686 

371789 

9735 

7426 

76978 

542963 

3869 

752784 

(27) 

(28) 

(29) 

(30) 

476583 

9759869 

898488 

5492284 

369868 

8387698 

764257 

9785746 

878725 

4555966 

701480 

9637478 

766958 

1804795 

925895 

4476204 

832777 

8468872 

699054 

6544058 

794489 

6804187 

854606 

9669733 

468985 

7557688 

988987 

1368572 

739849 

5687899 

504469 

2943489 

747494 

8848786 

566587 

6918394 

836366 

9579665 

677876 

31647o8 

669118 

3695467 

757769 

5819404 

577669 

6086164 

515987 

8578308 

8402326 

492834 

7533579 

13 


14  NEW  ADVANCED  ARITHMETIC. 

How  were  these  numbers  written  for  addition?  Wliy? 
With  what  cohimn  was  the  addition  begun  in  each  case? 
"Why?  W^hen  the  sum  of  the  numbers  expressed  in  any 
column  exceeded  nine,  what  was  done? 

35.  FORMAL    STATEMENT. 

Write  the  numbers  to  be  added  so  that  units  of  the  same 
order  shall  stand  in  the  satne  eolumum 

Jieo inning  tvith  the  lowest  order,  find  the  stim  of  the 
numbers  expressed  in  each  column*  If  this  sum  exceeds 
nine,  in  any  ease,  reduce  it  to  the  next  higher  denomina- 
tion, placing  the  remainder,  if  any,  under  the  column 
added,  and  adding  the  rediicetl  number  to  the  first  term  in 
the  next  higher  order. 

36.  TEST    OF    ACCURACY. 

1.  Add  the  numbers  expressed  in  each  column,  in  the  re- 
verse order. 

2,  Separate  the  problem  into  two  or  more  problems  and 
then  unite  the  several  results. 

What  must  be  true  of  numbers  in  order  that  they  may  be 
united? 

What  is  the  denomination  of  the  sum? 

37.  How  many  mills  are  there  in  one  cent?  cents  in  one 
dime?  dimes  in  one  dollar? 

38.     TABLE    OF    FEDERAL    MONEY. 

10  mills  make  1  cent. 
10  cents  make  1  dime. 
10  (limes  make  1  dollar. 

39.  In  writing  Federal  Money,  separate  the  order  of  dimes 
from  the  order  of  dollars  by  a  decimal  point.  126  dollars,  4 
dimes,  7  cents,  and  6  mills  is  written,  $126,476.  It  is  usually 
read,  126  dollars,  47  cents,  6  mills. 


ADDITION.  15 

40.  Write  the  following: 

1.  83  dollars,  27  cents,  4  mills. 

2.  59  dollars,  20  cents,  1  mill. 

3.  73  dollars,  5  cents,  2  mills. 

4.  148  dollars,  2  dimes,  9  mills. 

5.  300  dollars,  7  mills. 
Read  the  above  numbers, 

(1)  as  dollars,  dimes,  cents,  and  mills. 

(2)  as  dollars,  cents,  and  mills. 

41.  Find  the  sum  of  the  following  numbers: 

125  dollars,  26  cents,  8  mills. 
64  dollars,  33  cents,  4  mills. 
278  dollars,  5  cents. 
471  dollars,  6  mills. 
312  dollars,  59  cents,  7  mills. 

42.  Tell  how  the  numbers  are  written  for  addition.  Tell 
where  the  addition  should  begin.  Describe  the  process, 
using  a  form  similar  to  that  on  page  14. 

43.  Write,  read,  and  add  the  following : 

1.  $48,041,  863.247,  $146.28,  $276,007,  $160,406. 

2.  $361.79,  $483,062,   $583,802,  $1272.84,   $2169.176. 

3.  $2678.145,  $5684.297,  $462.01,  $5000.36,  $790.46, 
$579,614. 

4.  $2638.95,  $5406.63,  $2384.25,  376.52,  $857.35, 
$96834.67,  $3790.48. 

5.  $3762.05,  $67452.84,  $3568.90,  $3.5.70,  $49.32,  $5.83, 
$23.71. 

6.  $4.21,  $3.85,  $9.63,  $85.16,  $128.95,  $673.70, 
$2895..30,  $853.60. 

7.  $893.40,  $6.87,  $3708.90,  $7570.00,  $0.75,  54603.55, 
$3780.25. 


16  NEW  ADVANCED  ARITHMETIC. 

8.  S5.25,  SG0.70,  $375.08,  895.80,  §3617.50,  8817.46, 
?3064.38,  80.85. 

9.  83768.60,  8479.00,  86328.50,  82.91,  8325.75,  89.00, 
$754.08,  835.10. 

10.  87000.00,  8215.80,  8725.60,  850.50,  81.87,  86384.50, 
$4536.40. 

44.  How  many  gills  are  there  in  one  pint?  pints  in  one 
quart?  quarts  in  one  gallon?  These  are  used  in  measuring 
what? 

Write  the  table  of  Liquid  Measure. 

The  abbreviation  for  gallon  is  gal. ;  for  quart  is  qt.;  for  pint  is 
pt. ;  for  gill  is  gi. 

1.    Add :  5  gal.  2  qt.  1  pt.  3  gi. 

4    "    3    "  0    "  2   " 

7    "    1    "  1    "  2   " 

12     "    2    "  1    "  3  " 

14    "    0    "  0   "  1   " 


How  are  these  numbers  wi-itten  for  addition?  Where  does  the 
addition  begin  ?  Describe  the  process  as  you  did  in  Federal 
Money. 

2.    Add:  6  gal.  1  qt,  1  pt.  3  gi. 

4    "    0    "   1    "    2  '* 

12    "    3    "   0   "    1  " 

27    '^    2    "    1    "    2  '* 

36    "    3    "  0   "    0  " 


3.  Add  8  gal.  3  qt.  1  pt.  3  gi. ;  8  gal.  2  qt.  1  pt.  2  gi. ; 
4  gal.  1  qt.  1  pt. ;  1  gal.  1  pt.  1  gi.  ;  6  gal.  3  qt.  3  gi.  ; 
6  gal.    2  gi. 

45.  How  many  quarts  are  there  in  one  peek?  How  many 
pecks  in  one  bushel? 

Write  the  table  of  Dry  Measure. 


ADDITION. 


17 


1.   Add; 


5  bu.  3  pk.  6  qt, 

6  '^    2    '^  4 
9    "    1    "  6 

28    ''    0    "  7 

16    "    2    "  5 

4    "    3    "  2 


2.  Add  4  bu.  1  pk. ;  7  bu.  3  pk.  2  qt. ;  12  bu.  2  pk.  6  qt. ; 
19  bu.  1  pk.;  26  bu.  3  pk. ;   14  bu.  ;   18  bu.  1  pk.  ;  2  pk. 

3.  Add  21  bu.  1  pk. ;  36  bu.  2  pk. ;  19  bu.  3  pk. ;  2  pk. ; 
12  bu.  3  pk. ;   6  bu.  4  qt. 

46.  Write  the  following  problems  very  carefully  on  paper, 
slate,  or  blackboard,  and  solve : 

1.  824  +  69  +  703  +  9208  +  29607  =  ? 

2.  65  +  20007  +  893  +  566  +  15869  +  587  =  ? 

3.  5607  +  20189  +  46827  +  463912  +  51872  +  56928  + 
324501  +  873  =  ? 

4.  70128  4-  58694  +  79106  +  436912  +  586107  +  371009 
+  400106  =3  ? 

5.  4007  +  93281  +  56185  +  47594  +  508069  +  724378  + 
563128  =  ? 

6.  Paid  $2480  for  a  farm,  S268  for  a  span  of  horses,  |65 
for  a  wagon,  $32  for  a  set  of  harness,  S18  for  a  plough, 
$124  for  a  mowing-machine,  and  $384  for  other  utensils. 
What  was  the  entire  cost? 

7.  The  area  of  Maine  is  33,040  square  miles ;  of  New 
Hampshire,  9,305;  of  Vermont,  9,565;  of  Massachusetts, 
8,315  :  of  Rhode  Island,  1 ,250 ;  of  Connecticut,  4,990.  AVhat 
is  the  entire  area  of  the  New  England  States  ? 

8.  The  area  of  New  York  is  49,170  square  miles ;  of  New 
Jersey,  7,815;  of  Pennsylvania,  45,215;  of  Delaware,  2,050; 
of  Maryland,  12,210;  of  District  of  Columbia,  70;  of  Vir- 
ginia, 42,450;  of  AVest  Virginia,  24,780.  What  is  the  area 
of  the  Middle  States? 


18'  NEW  ADVANCED  ARITHMETIC. 

9.  The  area  of  North  Carolina  is  52,250  square  miles;  of 
South  Carolina,  30,570;  of  Georgia,  59,475;  of  Florida, 
68,680;  of  Tennessee,  42,050;  of  Alabama,  52,250;  of 
Mississippi,  46,810;  of  Louisiana,  48,720;  of  Texas, 
265,780;  of  Arkansas,  53,850;  of  Indian  Territory,  64,690. 
What  is  the  area  of  the  Southern  States? 

10.  What  is  the  united  area  of  the  New  England^  Middle, 
and  Southern  States? 

11.  On  Monday,  a  merchant  sold  3  gal.  2  qt.  1  pt.  of 
molasses ;   on  Tuesday,  5  gal.  3  qt. ;  on  Wednesday,  4  gal. 

1  pt. ;  on  Thursday,   6  gal.  1   qt.  1  pt. ;   on  Priday,   2  gal. 

2  qt.  1  pt. ;    on  Saturday,  7  gal.  3  qt.  1  pt.     How  much  did 
he  sell  in  the  entire  week? 

12.  A  dealer  sold  to  A,  426  bu.  3  pk.  2  qt.  of  oats ;  to  B, 
329  bu.  3  pk.  5  qt. ;  to  C,  189  bu.  1  pk.  7  qt. ;  to  D, 
426  bu.  2  pk.  5  qt. ;  to  E,  562  bu.  3  pk.  6  qt.  How  much 
did  he  sell  to  all? 

13.  A  farmer  values  his  horses  at  $350,  his  cows  at  $275, 
his  sheep  at  $411.75,  his  hogs  at  $129.25,  and  his  poultry  at 
$27.25.     What  is  his  value  of  all? 

14.  From  New  York  to  Albany  is  143  miles;  Albany  to 
Suspension  Bridge,  304  miles ;  Suspension  Bridge  to  Detroit 
230  miles ;  Detroit  to  Chicago,  284  miles.  What  distance 
from  New  York  to  Chicago? 

15.  A  merchant  bought  at  one  time  224  barrels  of  flour 
for  $1,344  ;  at  another,  217  barrels  for  $1,193.50;  at  another, 
192  barrels  for  $1,056;  at  another,  486  barrels  for  $2,916. 
How  many  barrels  did  he  buy?     What  was  the  cost  of  all? 

16.  A  steamship  sailed  239  miles  each  day  for  three  days, 
and  227  miles  each  day  for  the  next  three  days.  How  far 
did  she  sail  in  the  six  days? 

17.  Add  .26.3,  .187,  2.08,  4.019,  16.008,  .563,  .472,  .008. 

18.  Add  26.0029,  .0086,  .0278,  43.0196,  217.  .3059, 
.0062,  .0487. 


ADDITION.  19 

19.  Add  .00869,  .0571,  36.426,  68.07956,  .00964,  .473, 
.86o2,  .17,  .0080. 

20.  Add  .000524,  .00524,  .0524,  .068397,  6.080576, 
12.008642,  99.0864,   .000748,  .429783. 

21.  A  man  gave  each  of  his  four  sons  S375  ;  he  gave  his 
daughter  as  much  as  he  gave  to  two  sons ;  and  to  his  wife 
as  much  as  he  gave  to  three  sons.  How  much  did  he  give 
to  all? 

22.  A  man  left  to  his  heirs  a  certain  quantity  of  land. 
To  the  first  he  gave  320  acres ;  to  the  second  as  much  as  to 
the  first,  and  80  acres  more ;  to  the  third  as  much  as  to  the 
second,  and  96  acres  more;  to  the  fourth  as  much  as  to 
all  of  the  others,  and  to  his  wife  as  much  as  to  the  first 
three.     How  many  acres  did  he  bequeath? 

23.  In  1893,  the  enrollment  in  the  public  schools  of  Illi- 
nois was  as  follows :  Number  of  male  pupils  enrolled  in 
graded  schools,  228,412;  number  of  females,  235,885; 
number  of  male  pupils  enrolled  in  ungraded  schools,  189,851 ; 
number  of  females,  171,937.  What  was  the  total  enrollment 
for  the  year? 

24.  For  the  same  year  the  number  of  male  teachers  in 
graded  schools  w^as  1,692;  of  female  teachers,  was  8,410; 
the  number  of  male  teachers  in  ungraded  schools  was  4,861 ; 
the  number  of  female  teachers  was  9,277.  What  was  the 
whole  number  of  teachers  employed? 

25-  For  the  same  year  the  amount  paid  to  male  teachers 
in  graded  schools  was  $1,337,360.50;  to  female  teachers, 
$1,168,903.82;  to  male  teachers  in  ungraded  schools, 
$4,272,782.46;  to  female  teachers,  $1,641,281.79.  What 
was  the  whole  amount  paid  to  teachers? 

26.  The  Permanent  School  Fund  of  Illinois  is  as  follows : 
School  Fund  Proper,  $613,362.96;  the  Surplus  Revenue 
Fund,  $335,592.32;  the  College  Fund,  $156,613.32;  the 
Seminary  Fund,  $59,838.72  ;  the  County  Funds,  $158,616.63 ; 


20 


NEW  ADVANCED  ARITHMETIC. 


the  Township  Funds,  $12,220,722.14;  the  University  Fund, 

$006,207.6-1.     What  is  the  entire  fund? 

27.    The  following  form  is  called  a  "■  Bill,"  or  "  Statement 

of  account." 

Columbus,  Ohio,  May  15,  1896. 

James  Watson 

To  Robert  S.  William*,  Dr. 


1 

10 

72 
60 

•  1, 

25 

1 

10 

4 

1  Fifth  Reader, 

1  Advanced  Arithmetic, 

1  English  Grammar, 

1  Geography, 

1  History  United  States, 

Received  payment, 

Robert  S.  Williams. 

28.  John  Jones,  of  Utica,  N.  Y.,  bought  of  Pixley  &  Co., 
September  12,  1895,  the  following  articles.  Prepare  the 
bill. 

1  suit  of  clothes,  $16.50;  3  pair  hose,  90-  ;  1  light  over- 
coat, $12;  6  pair  cuffs,  $1.35;  1  doz.  collars,  $2.30;  1  hat, 
$3.25;  1  umbrella,  $1.85.     Receipt  the  bill. 

29.  On  January  24,  1896,  Frank  Walton  bought  of 
William  Snow  of  Louisville,  Ky.,  1  barrel  flour,  $3,25;  3 
hams,  $3.83;  30  lbs.  sugar,  $1.45;  5  lbs.  coffee,  $1.85;  2 
dozen  oranges,  $0.55 :  6  cans  tomatoes,  $1.15.  Bill  not  paid. 
Make  the  statement. 

30.  Bill  of  Robert  Thompson,  Bloomington,  111.,  rendered 
to  James  Dixon,  May  12,  1896.  Items:  Turkish  lounge, 
$25.00;  center  table,  $14.50;  6  chairs,  $13.35;  rocker, 
$4.75;  chiffonier,  $18.75;  hall  tree,  $17.50. 

31.  Bill  of  Joseph  Stoner,  blacksmith,  rendered  to  Charles 
Smith,  December  10,  1895,  Albany,  N.  Y.  Items:  sharpen- 
ing 3  plows,  $1.20;  shoeing  team,  $2.40;  setting  tires, 
$1.65;  repairing  buggy,  $1.50;  babbitting  harvester,  $5.75. 
Receipt  the  bill. 


ADDITION.  21 

47.   The  following  device  is  often  used  bj  accountants; 

83457  26 

29063  21 

840521  25 

364307  35 

428630  32 

976538  27 


2725516 


For  the  result  read  all  of  the  last  result  and  the  right-hand 
figures  of  the  previous  results. 

1.  Find  the  value  of  the  U.  S.  coinage  of  1891  from  the 
following  statement:  double  eagles,  825,891,340;  eagles, 
$1,956,000;  half  eagles,  81,347,065  ;  quarter  eagles,  827,600; 
silver  dollars,  823,562,735;  half  dollars,  8100,300;  quarter 
dollars,  81,551,150;  dunes,  82,304,671.60;  nickels,  8841,- 
715.50;  cents,  8470,723.50. 

2.  Money  in  circulation  in  United  States,  Dec.  1,  1894, 
was  as  follows  :  gold  coin,  8465,789,187;  gold  certificates, 
858,925,899;  silver  dollars,  857,449,865;  minor  coins, 
861,606,967;  silver  certificates,  8332,317,084 ;  "Sherman" 
notes,  8124,574,906;  United  States  notes,  8276,910,489; 
currency  certificates,  857,135,000;  National  Bank  notes, 
$202,517,054.     What  was  the  total  amount  in  circulation? 

3.  Distances   along   the   Chicago  and  Alton  Railroad  — 

Chicago  to  Joliet,  37  miles;  Joliet  to  Bloomington,  89  miles; 
Bloomington  to  Springfield,  59  miles ;  Springfield  to  Alton, 
72  miles ;  Alton  to  St.  Louis,  26  miles.  What  is  the  total 
length  of  the  road? 

4.  Two  ships  meet  in  mid-ocean.  One  sails  416  mUes 
eastward  the  first  day,  386  miles  the  second  day,  and  369 
miles  the  third  day.  The  other  sails  396  miles  westward 
the  first  day,  278  the  second  day,  and  339  the  third  day. 
How  far  apart  are  they  at  the  end  of  the  third  day? 

3A 


22  NEW  ADVANCED  ARITHMETIC. 


SECTIOIS^    III. 

SUBTRACTION. 

48.  As  Addition  is  the  process  of  uniting  two  or  more  like 
numbers,  Subtraction  is  the  process  of  separating  a  number 
into  two  smaller  numbers.  As  such  facts  as  2  +  0  =  2  are 
not  counted  in  the  addition  table,  so  facts  like  2  —  0  =  2  are 
not  counted  in  the  subtraction  table. 

49.  The  following  are  the  81  primary  problems  in  subtrac- 
tion. Neither  accuracy  nor  rapidity  is  possible  uutU  the 
difference  between  the  numbers  in  each  pair  can  be  given 
with  readiness. 

2  3  3  4  4  4  5 
12             13             2             14 

5  5  5  6  6  6  6 

3  2  15  4  3  2 

6  7  7  7  7  7  7 
16              5              4              3              2  1 

8  8  8  8  8  8  8 

7  6  5  4  3  2  1 

9  9  9  9  9  9  9 

8  7  6  5  4  3  2 


9 

10 

10 

10 

10 

10 

10 

1 

9 

8 

7 

6 

5 

4 

10 

10 

10 

11 

11 

11 

11 

3 

2 

1 

9 

8 

7 

6 

SUBTRACTION.  23 


11 

11 

11 

11 

12 

12 

12 

5 

4 

3 

2 

9 

8 

7 

12 

12 

12 

12 

13 

13 

13 

6 

5 

4 

3 

9 

8 

7 

13 

13 

13 

14 

14 

14 

14 

6 

5 

4 

9 

8 

7 

6 

14 

15 

15 

15 

15 

16 

16 

5 

9 

8 

7 

6 

9 

8 

16  17  17  18 

7  9  8  _9 

50.  John  had  17  marbles  and  lost  8,  How  many  had  he 
left? 

How  many  objects  would  be  needed  to  illustrate  this 
problem  ?  Use  the  problem  to  test  the  accuracy  of  the  fol 
lowing  definitions : 

51.  Subtraction  is  the  process  of  separating  a  number  into 
two  parts,  one  of  which  is  given  for  the  purpose  of  finding 
the  other. 

52.  The  Minuend  is  a  number  that  is  to  be  separated  into 
two  parts,  one  of  which  is  given  for  the  purpose  of  finding 
the  other. 

53.  The  Subtrahend  is  the  given  part  of  the  minuend. 

54.  The  Remainder  or  Difference  is  the  required  part  of 
the  minuend. 

55.  The  sign  —  (minus)  when  placed  between  two  num- 
bers shoves  that  their  difference  is  to  be  found. 

If  the  minuend  is  dollars,  what  will  the  subtrahend  be? 
Why?     What  will  the  remainder  be?     Why? 

56-  The  minuend,  subtrahend,  and  remainder  are  like 
numbers. 


24  NEW  ADVANCED  ARITHMETIC. 

John  had  17  marbles  and  James  had  12.  How  many 
more  had  John  than  James? 

How  many  marbles  would  be  needed  to  illustrate  this 
problem?     Will  the  definition  given  include  this  problem? 

57.  Problems  in  subtraction  assume  these  two  forms. 
When  objects  are  employed  they  are  reducible  to  the  first 
form  (see  Art.  50),  since  the  12  marbles  that  James  had 
indicate  the  size  of  one  of  the  two  parts  into  which  John's 
are  to  be  separated. 

58.  A  problem  in  subtraction  is  in  its  simplest  form 
when  each  term  in  the  minuend  equals  or  exceeds  the  cor- 
responding term  in  the  subtrahend. 

Illustration. 
8462     Minuend. 
5341     Subtrahend. 

59.  Problems  are  not  generally  in  this  form,  but  they 
must  be  made  to  assume  it  before  the  subtraction  can  be 
performed. 

Problem. 

721 

564 

Is  this  problem  in  its  simplest  form  ?  If  not,  show  why 
it  is  not. 

60.     EXPLANATION.  - 

1.  Since  the  ones'  term  of  the  minuend  is  less  than  the 
corresponding  term  of  the  subtrahend,  one  of  the  tens  in 
the  minuend  may  be  reduced  to  ones  and  added  to  the 
ones'  term.  10  ones  plus  1  equal  11  ones.  11  minus  4 
equals  7. 

2.  Since  the  tens'  term  of  the  minuend  is  less  than  the 
corresponding  term  of  the  subtrahend,  one  of  the  hundreds 
in  the  minuend  may  be  reduced  to  tens  and  added  to  the 
tens'  term.  10  tens  plus  1  ten  equal  11  tens.  11  tens  minus 
6  tens  eoual  5  tens. 


SUBTRACTION.  25 

3.    6  hundreds  minus  5  hundreds  equal  1  hundred. 

61.  Since  the  minuend  has  been  separated  into  the  sub- 
trahend and  remainder  their  sum  must  equal  the  minuend. 
Hence, 

To  prove  a  probletn  in  subtraction,  find  the  sum  of  the 
subtraJiend  and  remainder;  if  it  equals  the  minuendf 
the  ivork  is  correct, 

62.  Since  the  problem  given  is  like  all  problems  in  sub- 
traction, all  may  be  solved  as  this  has  been.  Hence,  we 
may  make  the  following 

RULE  FOR  SUBTRACTION. 

Write  the  subtrahend  below  the  minuend,  tvith  units 
of  the  same  order  in  the  same  column. 

Beginning  tvith  the  lowest  denomination,  subtract  each 
tenn  of  the  subtraJiend  from  the  corresponding  term  of 
the  minuend.  If  any  term  of  the  subtrahend  exceeds 
the  corresponding  term  of  the  minuend,  increase  the 
smaller  term  by  one  of  the  next  higher,  and  proceed  as 
before. 

63.     EXAMPLES  FOR  PRACTICE. 

1.    $24,685  6.    5444454 

13.574  4567956 


2.  $369.72 

126.41 

3.  $48,567 

26.785 


4.  $584.32 
296.48 


5.  $3557.888 
1899.899 


7. 

4222332 

2789679 

8. 

26689888 

16789899 

9. 

92820 

63574 

10. 

102875 

63986 

26 


NEW  ADVANCED   ARITHMETIC. 


11. 


12. 


13. 


14. 


15. 


16. 


17. 


18. 


19. 


20. 

Practice 
with  great 

31. 
32. 
33. 
34. 
35. 
36. 


146238 
87159 

196205 
159438 

2666776 
1789789 

611345566 

235845897 

833344565 
436789768 

746643455 
366799687 

82345533 
49455769 

923334454 

583584889 

3445444554 
1575678567 

5112556 
2686897 


21.  14455677 
4857898 


22.  84557666 
59698979 


23.  423445575 
259576976 


24.  21234566 
17457' 


25.  170308444 
24154456 


26.  8800337.005 

347942.462 

27.  40080222 
28848567 


28.  847000221 
512356345 


29.  $85200620.05 

40065010.23 

30.  403201058 
250042006 


Remainder. 

9 


on  these  30  problems  until  you  can  read  the  results 
rapidity. 

Minuend.  Subtrahend. 

824  63 

978  ?  467 

?  826  279 

1276  ?  638 

?  745  12864 

24671  ?  18983 


SUBTRACTION.  27 

How  find  subtrahend  when  fninuend  and  remainder  are 
given?     Wliy?     Make  a  rule. 

How  find  minuend  when  subtrahend  and  remainder  are 
given  ?     Why  ?     Make  a  rule. 

37.  60241—42374  +  26082  =  ? 

38.  280621  +  460082-31892  —  42671=? 

39.  529706-31793-64802  +  5729=? 

40.  A  merchant  bought  324  bu.  2  pk.  of  oats,  and  sold  to 
A  59  bu.  3  pk.  and  to  B  123  bu.  1  pk.  How  much  did  he 
have  left? 

41.  A  farmer  had  861  acres  of  land,  and  gave  294  acres 
to  his  son.     How  many  acres  did  he  have  left? 

42.  From  a  barrel  containing  37  gal.  3  qt.  1  pt.  and  3  gi. 
of  vinegar,  a  merchant  sold  to  A,  7  gal.  2  qt.  2  gi. ;  and  to 
B,  4  gal.  3  qt.  1  pt.  2  gi.  more  than  he  did  to  A.  How  much 
was  left  in  the  barrel  ? 

43.  C  and  D  were  350  miles  apart.  They  traveled  toward 
each  other,  C  at  the  rate  of  46  miles  a  day,  and  D  at  the 
rate  of  37  miles  a  day.  How  far  apart  were  they  at  the  end 
of  the  second  day?  Which  had  traveled  farther?  How 
much  ? 

44.  Four  men  together  owned  786  cattle.  E  owned  227, 
and  F,  339.  G  owned  as  many  less  than  E  as  F  did  more 
than  E.     How  many  did  H  own? 

45.  A  merchant  paid  $2,500  for  a  quantity  of  silk.  For 
other  dry  goods  he  paid  $265  more  than  he  did  for  the  silk. 
For  groceries  he  paid  $683  less  than  what  he  had  before 
expended.  To  one  customer  he  sold  goods  amounting  to 
$5,280  ;  to  another,  $325  less  than  half  as  much  ;  to  another, 
$2,895  less  than  to  the  first,  thus  disposing  of  all  of  his 
stock.     Did  he  gain  or  lose,  and  how  much? 

46.  One  cask  contains  39  gal.  1  qt.  1  pt.  2  gi.  of  alcohol, 
and  another  28  gal.  3  qt.  1  pt.  3  gi.  How  much  more  does 
the  first  contain  than  the  second  ? 


28  NEW  ADVANCED  ARITHMETIC. 

47.  In  one  bin  there  are  873  bu.  2  pk.  5  qt.  of  com ;  in 
another,  698  bu,  3  pk.  7  qt.  How  much  less  in  the  second 
than  in  the  first? 

48.  A  has  $650;  B  has  825  less  than  half  as  much.  C 
has  Si 25  less  than  both  A  and  B.  D  has  §275  less  than  A, 
B,  and  C.     How  much  have  all  of  them? 

49.  A  and  B  start  from  the  same  place  at  the  same  time 
and  travel  on  the  same  road.  A  travels  at  the  rate  of  26 
miles  an  hour,  and  B  at  the  rate  of  39  miles  an  hour.  How 
far  apart  will  they  be  at  the  end  of  6  hours?  If  B  should 
then  stop,  and  A  should  travel  at  B's  rate,  how  long  would 
it  take  him  to  overtake  B? 

50.  From  A  to  B  is  492  miles;  from  B  to  C  is  half  as 
many  miles,  plus  42.  How  find  the  difference  in  the  two 
distances  ?     How  many  miles  from  A  to  C  ? 

51.  Three  persons  bought  a  mill  valued  at  $25642.  The 
first  paid  S6743.25;  the  second  t«'ice  as  much;  and  the 
third,  the  remainder.     How  much  did  the  third  pay? 

52.  At  one  time  I  spent  ^  of  a  dollar ;  at  another,  |  of  a 
dollar ;  and  at  another,  |  of  a  dollar.  If  I  had  but  a  dollar 
at  the  beginning,  what  part  of  it  had  I  left?  How  many 
cents  ? 

53.  How  many  more  days  in  the  months  of  March,  April, 
May,  and  June,  counted  together,  than  in  the  months  of 
September  and  October? 

54.  "When  shopping,  a  lady  bought  ribbon  for  36  cents, 
lace  for  $1.48,  gloves  for  81.75,  and  velvet  for  81.27.  She 
gave  in  payment  a  five-dollar  bill,  and  received  her  change 
in  nickels  and  cents.     How  many  nickels  did  she  receive? 

55.  8.27-6.52  =  ? 

56.  264.008  -  79.169  =  ? 

57.  .80641  -  .27835  =  ? 

58.  .01803  —  .00657  =  ? 

59.  2.061  -  .8934  =  ? 


SUBTRACTION.  29 

60.  7.006521  -  .009730  =  ? 

61.  4.069  +  72.0083  -  10.15328  =  ? 

62.  .08031  +  .2483  +  .005687  -  .0148  -  00693  =  ? 

63.  263.094  -  172.86  +  52.0048  -  ? 

64.  $821,054  +  $63,006  -  $279,838  -  $346,765  =  ? 

65.  Bought  several  articles  costing  respectively  63;^,  89)?!, 
48?',  $1.38,  $2.76,  $4.75.  Gave  merchaut  a  $20  bill.  What 
ohaoge  should  I  receive  ? 

66.  How  many  years  after  the  discovery  of  America  was 
Washington  born?  How  old  was  Washington  at  the  time  of 
the  Declaration  of  Independence  ?  When  he  was  first  inau- 
gurated as  president?     (Omit  months  and  days.) 

67.  Lincoln  was  born  how  many  years  after  Washington? 
How  old  was  he  at  the  time  of  his  death?  If  still  living, 
how  old  would  he  be?     (Omit  months  and  days.) 

68.  How  many  years  from  the  battle  of  Bunker  Hill  to  the 
attack  on  Fort  Sumter?  From  the  surrender  at  Yorktown 
to  Lee's  surrender?  From  the  battle  of  Waterloo  to  the 
battle  of  Gettysburg? 

69.  The  National  Debt  of  U.  S.,  on  July  1,  1870,  was 
$2480672427.81.  On  July  1,  1894,  it  was  $1632253636.68. 
How  much  had  it  diminished  in  24  years  ? 

70.  The  area  of  France  is  204092  square  miles.  That  of 
Great  Britain  and  Ireland  is  120979  square  miles.  What  is 
the  difference  of  their  areas? 

Which  of  the  States  of  the  American  Union  is  larger  than 
either? 

It  is  how  much  larger  than  France  ?  Than  Great  Britain 
and  Ireland? 

Which  is  larger,  France  or  California?     How  much? 

Great  Britain  and  Ireland  or  California?     How  much? 

71.  How  many  years  ago  was  the  Declaration  of  Inde- 
pendence signed  ?  How  many  years  from  the  signing  of  the 
Declaration  of  Independence  to  the  close  of  the  Civil  War? 


80 


NEW  ADVANCED   ARITHMETIC. 


AREAS   OF   STATES    AND  TERRITORIES. 


NEW    ENGLAND. 


Maine, 

33,040. 

Vt.,            9,565. 

R.  I., 

1,2.50. 

N.  H., 

9,305. 

Mass.,       8,315. 

Conn., 

4,990. 

MIDDLE   ATLANTIC    GROUP. 

N.Y., 

49,170. 

Del.,           2,050. 

Va., 

42,450. 

N.  J., 

7,815. 

Md.,        12,210. 

W.  Va. 

,  24,780. 

Penn., 

45,215. 

D.  C,             70. 
COTTON    STATES. 

/ 

N.  C, 

52,250. 

Ala.,        52,250. 

Texas, 

265,780. 

S.  C, 

30,570. 

Miss.,      46,810. 

Ark., 

53,850. 

Ga. 

59,475. 

La.,         48,720. 

Teun., 

42,050. 

Fla. 

58,680. 

CENTRAL    STATES. 

Ky., 

40,400. 

111.,           56,650. 

Wis., 

56,040. 

Ohio, 

41,060. 

Mo,        69,415. 

Minn., 

83,365. 

lud., 

36,350. 

Mich.,     58,915. 
THE    GREAT    PLAIN. 

Iowa, 

56,025. 

N.  D., 

70,795. 

Neb.,       77,510. 

Okla., 

39,030. 

S.  D., 

77,650. 

Kan.,      82,080. 
MOUNTAIN    STATES. 

Ind.  T. 

,  31,400. 

Mont., 

146,080. 

Colo.,    103,925. 

N.  M., 

122,580. 

Idaho, 

84,800. 

Utah,       84,970. 

Ariz., 

113,020. 

Wy., 

97,890. 

Nev.,     110,700. 

Cal.,      158,360. 


PACIFIC   STATES. 

Oreg.       96,030. 

Alaska,  531,409. 


Wash.,    69,180. 


SUBTRACTION.  31 

Find  difference  in  area  between  — 

71.  Maine  and  the  rest  of  New  England. 

72.  Texas  and  the  other  Gulf  States. 

73.  New  England  and  Illinois. 

74.  The  Central  States  and  the  Cotton  States. 

75.  The  Pacific  States  and  the  Middle  Atlantic  Group. 

76.  The  Central  States  and  the  Great  Plain. 

^  77.   The  five  States  on  the  east  bank  of  the  Mississippi 
and  the  three  States  on  the  west  bank. 

78.  How  many  States  each  equal  to  Illinois  can  be  cut 
out  of  Texas  ? 

79.  Find  the  total  area  of  all  the  States  east  of  the  Mis- 
sissippi River,  excluding  Minnesota  and  Louisiana. 

80.  Find  the  total  area  of  the  original  thirteen  States. 

81.  A  man  expended  810,564  for  4  tracts  of  land.  For 
the  first  he  paid  Sl,9G8.50;  for  the  second,  $2,680;  for  the 
thu-d,  $3,127.50.     What  did  he  pay  for  the  fourth? 

82.  The  sum  of  the  areas  of  Maine,  Ky.,  Md.,  Penn.,  and 
a  fifth  State  is  189,072  square  miles.  What  is  the  area  of 
the  fifth  State?     Which  is  it? 

83.  A  man  bought  three  buildings.  He  paid  for  the  first 
$7,846;  for  the  second,  82,875  more;  for  the  thu-d,  83,182 
less  than  for  the  second.  He  put  in  as  part  payment  a  farm 
for  82,125,  and  paid  the  remainder  in  cash.  What  was  his 
cash  payment? 

Latitude  and  longitude  are  measured  in  degrees,  minutes,  and 
seconds.  Sixty  seconds  (marked")  make  a  minute.  Sixty  min- 
utes (marked  ' )  make  a  degree  (marked  °  ). 

84.  New  York  is  in  74  °  3  "  west  longitude  and  Boston 
71°  3'  30"  west  longitude.  Boston  is  how  far  east  of  New 
York? 

85.  Chicago  is  in  87°  35'  west  longitude.  How  far  is  it 
west  of  N.  Y.  ?     Of  Boston  ? 


32  NEW  ADVANCED  ARITHMETIC. 

86.  Albany  is  298  miles  east  of  Buffalo  aud  Chicago  is 
589  west  of  Buffalo.  AVhat  is  the  distance  from  Albany  to 
Chicago  ? 

87.  Berlin  is  13°  23' 43"  E.  and  New  Orleans  90°  3' 28" 
W. ;  what  is  the  difference  of  their  longitudes? 

88.  Boston  is  in  42°  21'  24"  N.  latitude.  The  latitude 
of  New  York  is  40°  42'  43"  N.  Boston  is  how  much 
farther  north  than  New  York? 

89.  London  is  in  latitude  51°  30'  48"  N.  It  is  how 
much  farther  north  than  Boston? 

90.  New  Orleans  is  29°  57'  N.  and  Rio  Janeiro  22°  54' 
S.     What  is  their  difference  in  latitude? 

91.  Find  the  time  from  July  6,  1888,  to  Sept.  10,  1896. 

92.  Find  the  time  from  March  12,  1889,  to  Oct.  18,  1895. 

93.  Find  the  time  from  June  15,  1887,  to  April  5,  1897. 

Note.  How  many  jears  from  June  15,  1887, to  June  15,  1896?  How 
many  months  from  June  In,  1896,  to  March  15,  189/  ?  How  many  days 
lu  March  after  the  15th  ?     To  these  add  the  5  days  in  April. 

94.  Find  the  time  from  Aug.  21,  1890,  to  May  16,  1897, 

95.  Find  the  time  from  Sept.  12,  1891,  to  Dec.  25,  1897. 

96.  Find  the  time  from  Oct.  28,  1886,  to  June  19,  1895. 


MUL  TIP  Lie  A  TION.  33 


SECTioivr  ly. 

MULTIPLICATION. 

64.  What  is  the  sum  of  5  and  5  ?  "What  is  the  smu  of  two 
5's?  What  is  the  sum  of  8  and  8  and  8?  What  is  the  sum 
of  three  8's?  AVhat  is  the  sum  of  five  6's?  of  four  9's?  of 
seven  lO's?  This  will  suggest  the  manner  in  which  the  mul- 
tiplication table  is  built  up.  As  commonly  used  it  consists 
of  sums  formed  by  repeating  the  nine  primary  numbers  up  to 
nine  of  each. 

65.  Multiplication  is  a  short  method  of  finding  the  sum  of 
tV70  or  more  equal  numbers. 

66.  The  Multiplicand  is  one  of  the  two  or  more  equal  num- 
bers that  are  to  be  united. 

67.  The  Multiplier  is  the  number  of  equal  numbers  that  are 
to  be  united. 

68.  The  Product  is  the  sum  of  two  or  more  equal  numbers 
that  have  been  united  by  multiplication. 

69.  The  sign  of  multiplication  is  an  oblique  cross.  When 
the  multiplier  comes  first,  the  sign  is  read  times  When  the 
multiplicand  precedes,  the  sign  is  read  multiplied  by. 

70.  Numbers  are  spoken  of  as  abstract  or  concrete.  This 
distinction  is  of  little  value  except  in  Multiplication  and 
Division.  A  number  of  named  objects,  as  6  books,  is  called 
a  concrete  number.  A  number  whose  unit  is  not  named,  as 
7,  or  a  number  of  numbers,  as  5  nines,  is  called  abstract. 
Accordingly, 

1.  The  Multiplicand  may  be  abstract  or  concrete. 

2.  The  Multiplier  is  abstract. 

3.  The  Product  is  like  the  Multiplicand. 


84 


NEW  ADVANCED  ARITHMETIC. 


71.  Multiplication  Table. 


1 

2 

3 

4 

5     6,7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12  ;  14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

23 

32 

36 

40 

44 

48 

6 

10 

15 

20 

Z5 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

66 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63  ■ 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 

66 

77 

88 

99 

108 

110 

121 

132 

12 

24 

3G 

48 

CO 

72 

84 

90 

120 

132 

144 

72.      EXAMPLES    FOR    PRACTICE. 

These  problems  and  others  like  them  should  be  used  until  the  results 
can  be  spoken  with  great  rapidity. 


1.  1348G9725 
2 


2.  843512769 
3 


3.  4G9 1058327 
4 


5.  7964031825 

6 

6.  6324507198 

7 

7.  31924065708 


4.    9601475328 
5 


8.    2573960814 
9 


MULTIPLICATIOX.  .^  35 

Go  through  these  problems,  giving  the  product  of  the 
multiplier  and  each  term  of  the  multiplicand  "v\'ithout  regard 
to  the  denomination  of  the  product. 

Repeat  the  operations,  naming  the  denomination  of  each 
product. 

Repeat  the  operations,  naming  the  denomination  of  each 
product,  reducing  it  to  the  next  higher  order  when  it  exceeds 
nine,  and  giving  the  denomination  of  the  remainder  and  of 
the  reduced  number  in  each  case. 

73.  "When  the  product  in  any  case  does  not  exceed  nme, 
where  should  it  be  -written  ?  "When  the  product  exceeds  nine, 
where  should  the  remainder  be  written?  What  should  be 
done  with  the  reduced  number? 

74.  Since  the  method  employed  in  these  cases  is  applica- 
ble iu  all  similar  cases,  we  may  make  the  following  rule  for 
multiplication  when  the  multiplier  is  less  than  10. 

RULE. 
Write  the  multiplier  under  the  lotvest  term  of  the  tniiU 
tiplicand.  MuUiply  this  term  of  the  imtltiplicand  by  the 
multiplier.  If  the  product  is  less  than  lO,  j^lace  it  tinder 
the  term  multiplied.  If  the  product  is  more  than  lO, 
reduce  it  to  the  next  higher  denomination,  placing  the  re- 
mainder, if  there  should  be  one,  under  the  term  mtilti- 
plied,  and  adding  the  reduced  number  to  the  product  of  the 
multiplier  and   the  next  higher  term  of  the  multiplicand, 

75.  ]Multiplication  by  tens,  hundreds,  etc. 

Illustrative  Example.        8364 

22 


16728 
16728 

184008 


76.  In  this  problem  the  tens'  term  of  the  multiplier  is  how 
many  times  the  units'  term?  The  product  of  each  tenu  of 
the  multiplicand  and  the  tens'  term  of  the  multiplier  is  how 


36  NEW  ADVANCED  ARITHMETIC. 

many  times  the  product  of  the  same  terms  and  the  units* 
term  of  the  multiplier?  Where,  then,  should  these  results 
be  written? 

77.  Since  a  figure  standing  in  tens'  order  expresses  a 
number  ten  times  as  great  as  if  standing  in  units'  order,  the 
products  obtained  by  using  the  tens'  term  of  the  multiplier 
will  belong  one  order  higher  than  the  terms  multiplied. 
Hence, 

To  multiply  by  tens,  proceed  as  before,  placing  the  results 
one  order  to  the  left  of  the  term  multiplied. 

78.  Apply  the  same  reasoning  to  the  hundreds',  thou- 
sands', etc.,  terms  of  the  multiplier,  and  make  a  statement 
for  each. 

79.     GENERAL    RULE. 

Write  the  tniiltiplier  tinder  the  multiplicand. 

Beginning  with  the  units'  term  of  the  multiplier,  mul- 
tiply the  multiplicand  by  each  term  of  the  multiplier, 
successively,  placing  the  right-hand  figtire  of  each  partial 
product  under  the  term  by  which  it  was  obtained. 

Unite  the  several  partial  products. 

80.     PROOFS. 

1.  Use  the  multiplicand  as  the  tmiltiplier. 

2.  Divide  the  product  by  either  of  its  factors.  The  qug^ 
tient  will  be  the  other  factor,  if  the  work  is  correct. 

Note.    Proof  2  cannot  be  used  until  after  a  study  of  division. 
81.    Illustrative  Analysis.     Multiply  826  by  352. 

1.  We  are  to  unite  352  826's.  "We  first  unite  2  826's,  then  50  826's, 
then  300  826's,  and  then  unite  these  several  products. 

Note.     826's  is  read  eight  hundred  twenty-sixes. 

2.  2  times  826  one."  are  1652  ones. 

3.  50  times  826  are  5  times  ten  times  826.  Ten  times  826  are  826  tens. 
Since  5  times  826  tons  are  tens,  the  ric;ht-hand  figure  of  tliis  partial  pro- 
duct will  be  written  under  the  tens'  figure  of  the  multiplicand. 


MULTIPLICATION.  37 

4.  300  times  826  are  3  times  100  times  826.  100  times  826  are  826  hun- 
dreds. Siuce  3  times  826  Imndreds  are  hundreds,  the  right-hand  figure  of 
this  partial  product  will  be  written  under  the  hundreds'  figure  of  the  mul- 
tiplicand. 

5-    Uniting  the  partial  products  gives  the  complete  product. 

82.      EXAMPLES    FOR    PRACTICE. 

1.  Multiply  864  by  24;  by  35,  by  o8 ;  by  79 

2.  Multiply  978  by  61 ;  by  83 ;  by  96  ;  by  89. 

3.  Multiply  4625  by  189  ;  by  265  ;  by  374. 

4.  Multiply  3718  by  264  ;  by  357;  by  819. 

5.  Multiply  24689  by  345  ;  by  678  ;  by  921, 

6.  Multiply  13576  by  987;  by  654;  by  321. 

7.  Multiply  90876  by  234  ;  by  567  ;  by  892. 

8.  Multiply  34005  by  306  ;  by  508  :  by  809. 

9.  874  X  81326. 

10.  596  X  493725. 

11.  2086  X  321068. 

12.  34708  X  97540016. 

13.  65413  X  2496038. 

83.  To  multiply  by  10,  100,  1000.  etc.,  annex  as  many 
zeros  to  the  right  of  the  multiplicand  as  there  are  in  the 
multiplier. 

Note.  If  the  decimal  point  is  used,  it  must  be  moved  as  many  orders 
to  the  right  as  there  are  zeros  in  the  multiplier,  and  the  vacant  orders 
m*st  be  filled  with  the  zeros. 

84.      PROBLEMS. 

1.  Multiply  7642  by  10;  by  100;  by  10000;  by  1000;  by 
100000 

2.  Multiply  246092  by  1000;  by  10;  by  10000;  by  100. 

3.  Multiply  9284  by  200 :  by  3000;  by  60  ;  by  70000. 

4.  Multiply  5681  by  2600 ;  by  87000 ;  by  24600 ;  by  360. 

4A 


38  NEW  ADVANCED  ARITHMETIC. 

85.      GENERAL    PROBLEMS. 

1.  What  is  the  cost  of  5  cords  of  wood  at  86  a  cord? 

Analysis.  Since  each  cord  cost  $6,  5  cords  cost  5  sixes  of  dollars  (or  5 
times  S6),  which  are  $30. 

2.  What  is  the  cost  of  236  acres  of  land  at  $72  an  acre? 
Analyze  as  above. 

Note.  In  such  problems  as  number  2,  where  the  multiplicand  is  smaller 
than  the  multiplier,  the  order  of  factors  may  be  changed  by  the  following 
analysis : 

If  the  land  cost  $1  an  acre,  236  acres  would  cost  $23G.  Since  the  laud 
cost  $72  an  acre,  236  acres  cost  72  times  S236.  Use  this  form  of  analysis 
until  it  cau  be  employed  easily  and  rapidly. 

3.  If  a  man  travel  at  the  rate  of  46  miles  a  day,  how 
many  miles  will  he  travel  in  27  days? 

4.  James  earned  S264;  John,  $432;  William,  twice  as 
much  as  both  James  and  John;  Henry,  three  times  as  much 
as  the  difference  between  William's  earnings  and  John's 
earnings.     What  did  all  earn? 

5.  Thomas  bought  4  gal.  3  qt.  1  pt.  2  gi=  of  milk ,  Eeuben 
bought  5  times  as  much.     How  much  did  both  buy? 

6.  If  16  men  can  do  a  piece  of  work  in  12  days,  in  how 
many  days  can  one  man  do  it? 

7c  If  a  railway  train  run  42  miles  an  hour,  how  many 
miles  will  it  run  in  678  hours? 

8.  A  locomotive  division  is  126  miles  long.  How  many 
miles  does  an  engineer  ride  in  July,  if  he  makes  a  trip  every 
day? 

9.  A  merchant  bought  7  loads  of  oats,  each  containing  63 
bu.  2  pk.  o  qt.  How  many  did  he  buy?  What  did  they  cost 
him,  at  30  cents  a  bushel  ? 

10.  The  distance  from  Bloomington  to  Chicago  is  126 
miles.  How  many  feet  of  wire  are  there  in  7  telegraph  lines 
connecting  the  two  cities? 


MUL  TIP  Lie  A  TION.  39 

U.  A  man  had  $3,000.  He  bought  6  horses  at  $125  each, 
5  cows  at  $57  each,  2  wagons  at  $48  each,  and  40  acres  of 
land  at  $40  an  acre.     How  much  money  had  he  left? 

12.  Multiply  the  sum  of  826  and  439  by  twice  their  dif- 
ference. 

13.  Multiply  the  sum  of  I  X  729  and  8  X  563  by  9  times 
the  difference  between  23  X  48  and  65  x  76. 

86.     MULTIPLICATION    BY    FACTORS. 

Since  6  =  2  x  3,  6  times  any  number  =  2  x  S  times,  or 
3x2  times,  that  number  ^  hence,  if  I  wish  to  multiply  a 
number  by  6,  I  may  multiply  it  by  3  and  that  product  by  2, 
or  multiply  it  by  2  and  that  product  by  3. 

The  same  plan  may  be  followed  with  any  number  that 

can  be  factored. 

RULE. 
Separate  the  multiplier  into  two  or  more  factors,  mul- 
tiply hy  one  of  them,  the  resulting  product    by  a  second^ 
and  so  continue  until  all  of  the  factors  have  been  used. 
The  last  product  is  the  reQtiired  result. 

87.      PROBLEMS. 

1.  Multiply  456  by  15,  using  factors  of  15. 

Analysis.     Since  15  =  3  X  5,  15  times  456  =  3  times  5  times  456. 

5  times  456  =  2280;   3  times  2280  ==  6840. 

FORM. 

2.  5246X18=?  ^246 

3.  6792x25  =  ?  15738 

4.  24680  X  48  =  ?  6 

5.  56072X64  =  ?  ^^^-^ 

88.  Multiplication  by  numbers  slightly  less  than  a  power 
of  10. 

1.    869  X  99  =  ? 

Analysis.  If  the  multiplier  were  100,  the  product  would  be  86900. 
Since  the  multiplier  is  one  less  than  100,  the  product  is  once  869  less  than 
86900.     86900  -  869  =  86031. 


40  NEW  ADVANCED  ARITHMETIC. 

2.  Multiply  7824  by  999  ;  by  9999. 

3.  Multiply  862534   by  98;    by  998;    by  97;    by  997; 
by  9997.  , 

89.    Learn  the  following  table  of  aliquot  parts ; 


16|=  J  of  100. 

1G6|  =  J  of  1000. 

2|  r=  i  of  10. 

25    -lot  100. 

250    =  1  of  1000. 

5    =  J  of  10. 

50    =  ^  of  100. 

500    =  ^  of  1000. 

7^  =  1  of  10. 

75    =  1  of  100. 

750    =  f  of  1000. 

3i  =  ^  of  10. 

33i  =  iof  100. 

333i  =  i  of  1000. 

6|  =  f  of  10. 

66f  =  f  of  100. 

666|  =  1  of  1000. 

12^  -^oi  100. 

125    =  i  of  1000. 

371  =  1  of  100. 

375    =  1  of  1000. 

&2h  =  1  of  100. 

625    —  1  of  1000. 

8^=  1  of  10. 

83i  =  1  of  100. 

833i  =  f  of  1000. 

8|  =  |of  10. 

87^  =  1  of  100. 

875    =  1  of  1000. 

The  twelfths  and  ; 

sixteenths  of  100  and 

1000  may  be  added  for  more 

extended  work. 

Illustrative  Example. 

FORM. 

1-284 

1284 

X  37^=? 

128400 
1(3050 
48150 

Analysis.  37^^=1  of  100.  If  the  multiplier  were  100,  the  product 
would  be  128400.  If  the  nmltiplier  were  |  of  100,  tlie  product  would  be 
i  of  128400,  which  equals  16050.  Since  the  multiplier  is  |  of  100,  the 
product  is  3  times  16050,  which  equals  48150. 

90.      PROBLEMS. 

1.  Multiply  48464  by  62i;  by  12|;  by  625;  by  375; 
by  75. 

2.  Multiply  58647  by  33,\;  by  66§;  by  333^;  by  666§. 

3.  Multiply  86484  by  16|;  by  166§;  by  831^;  by  833^. 

4.  What  is  the  cost  of  24  dozens  of  eggs,  at  12^  cents 
a  dozen  ? 

Analysis.  At  one  dollar  a  dozen,  24  dozens  cost  $24.  Since  the  price 
13  12i  cents,  or  |  of  a  dollar  a  dozen,  24  dozens  cost  \  of  $24,  which 
equals  $3. 


MUL  :iPLICATION.  41 

5.  What  will  16§  lbs.  of  sugar  cost  at  6  cents  a  pound? 
25  lbs.?  33^  lbs.?  G6§  lbs.?  83^  lbs.? 

Analysis.     100  lbs.  cost  $6.    i  of  100  lbs.  cost  \  of  S6. 

6.  What  is  the  cost  of  324  objects  at  8^  cts.  each?  at 
in  cts.?  at  16§  cts.?  at  25  cts.?  at  33^  cts.?  at  37i  cts.? 
at  50  cts.  ?  at  75  cts.  ? 

7.  Find  the  cost  of  625  objects  at  16|  cts.;  at  25  cts.; 
at  50  cts. ;   at  66^  cts. ;  at  40  cts. 

8.  Find  the  cost  of  824  objects  at  33^  cts. ;  at  62^  cts. ; 
at  37i  cts. ;  at  83^  cts. 

9.  Find  the  cost  of  Q2b  horses  at  $83^-  each;  at  §87^ 
each ;  at  $62i  each. 

10.  Find  the  cost  of  824  cattle  at  $37i  each;  at  §33^ 
each  ;  at  $50  each. 

91.     MISCELLANEOUS    PROBLEMS. 

1.  8  X  .024  =?     9  X  .0073  =?     27  X  -1694  =? 

2.  36x3.056=?    48X17.0835=?     59x24.16947=? 

3.  83  X  .24356  =?     92  x  .3607  =?     98x7.00869=? 

4.  Multiply  .086794  by  8  ;  by  9  ;  by  26  ;  by  37. 

5.  Multiply  365.432  by  45;  by  69;  by  138;  by  246. 

6.  Q>Q''^  X  7086957. 

7.  230000  x  569842. 

8.  998  X  47952. 

9.  (84625  +  53796)  X  (63824  —  21706). 

10.  (821  —  463  +  279)  X  (425  +  872  -  328). 

11.  Multiply  3  gal.  2  qt.  1  pt.  by  5  ;  by  8 ;  by  9  ;  by  12. 

12.  Multiply  7  yd.  2  ft.  8  in.  by  7 ;  by  10;  by  11. 

13.  Multiply  46  bu.  3  pk.  by  15  ;  by  24 ;  by  38  ;  by  49. 

14.  Bought  1 60  acres  of  land  at  $65  an  acre,  and  80  acres 
at  S75  an  acre.  Sold  120  acres  at  $80  an  acre  and  the 
remainder  at  $66.50  an  acre.     What  was  the  gain  or  loss? 


42  NEW  ADVANCED  ARITHMETIC. 

15.  A  dealer  bought  3  horses  at  S85  each ;  5  at  $96  each ; 
7  at  $124.50  each.  He  shipped  them  to  the  city,  the  freight 
averaging  $12.50  each.  He  sold  the  cheapest  at  $110.50 
each;  the  second  lot  at  $128  each,  and  the  third  lot  at  $164 
each.  If  his  personal  expenses  and  the  care  of  the  horses 
amounted  to  $32,  what  did  he  gain  by  the  transaction? 

16.  Put  the  following  problems  in  the  form  of  bills  with 
the  teacher  as  the  seller  and  yourself  as  purchaser : 

1.  12  yds.  calico  at  6]- f  ;  6  doz.  eggs  at  12^(/ ;  3i  lbs. 
coffee  at  42;*;  15  yds.  muslin  at  11  <^;  9  yds.  summer  silk 
at  86/  ;  3  pairs  shoes  at  $1.45  ;  2  bu.  potatoes  at  65;. 

2.  4  yds.  linen  at  40/;  2  pairs  gloves  at  $1.75;  6  yds. 
silk  at  $1.25;  15  yds.  sheeting  at  25^ ;  6  yds.  pillow-casing 
at  12i^';  8  yds.  towelling  at  24  ^ ;  ^  yd.  velvet  at  $1.25; 
6  handkerchiefs  at  25/  ;  2  waists  at  $1.25. 

17.  A  stone  falls  16  feet  the  first  second,  (16 +32^  feet 
the  second  second,  (16  +  32  +  32)  feet  the  third  second,  and 
so  on.     How  many  feet  will  it  fall  in  8  seconds? 

18.  Light  travels  186000  miles  a  second.  It  takes  498 
seconds  for  a  light  wave  to  pass  from  the  sun  to  the  earth. 
What  is  the  distance  ? 

19.  A  bushel  of  corn  in  the  ear  weighs  70  pounds ;  shelled, 
56  pounds.  How  many  pounds  of  cobs  in  a  crib  containing 
1800  bushels  of  ears? 

20.  Make  a  pendulum  by  fastening  a  split  bullet  to  a 
thread  30  inches  long.  Count  the  vibrations  in  a  second. 
How  many  vibrations  will  it  make  in  a  minute?  in  an  hour? 
in  a  day  ?  in  a  week  ? 

21.  A  car  is  loaded  with  49  steel  rails  32  feet  long  and 
weighing  78  pounds  to  the  yard.  The  weight  of  the  car  and 
its  load  is  62592  pounds.     AVhat  is  the  weight  of  the  car? 

22.  If  the  above  car  be  loaded  with  512  bushels  of  shelled 
corn,  what  will  the  car  and  its  load  weigh? 

23.  What  is  the  estimated  number  of  words  in  a  book 
containing  240  pages,  each  page  averaging  350  words? 


MULTIPLICA  TIOX. 
92.     SURFACE    MEASURE. 


43 


1.  This  figure  is  3  inches  long  and  2  inches  wide.  How 
many  square  inches  in  each  horizontal  row?  In  the  whole 
figure  ? 

2.  Draw  a  surface  5  inches  long  and  3  inches  wide. 
How  many  square  inches  in  each  row  ?  in  the  whole  figure  ? 

3.  This  page  is  5  inches  by  7  inches.  How  many  square 
inches  in  a  single  row  along  the  right-hand  margin?  How 
many  such  rows?     How  many  square  inches  in  the  page? 

Note.  Id  the  following  problems  the  measurements  should  be  made 
by  the  pupils,  with  tape-line,  yard-stick,  or  foot-rule.  The  nearest  integral 
number  of  units  should  be  taken.  Thus,  if  the  desk-top  in  the  uext  problem 
is  20f  inches  long,  it  should  be  taken  as  21  inches. 

4.  How  many  square  inches  in  your  desk-top?  in  all  the 
desk-tops  in  the  school-room? 

5.  How  many  square  inches  in  a  pane  in  your  nearest 
window?  in  all  the  panes?  in  all  the  windows? 

6.  How  many  square  feet  in  your  school- room  floor? 
square  inches? 


44  NEW  ADVANCED  ARITHMETIC. 

7.  In  a  well-lighted  school-room  the  floor-surface  is  not 
more  than  six  times  the  window  surface.  Is  your  room 
"well-lighted"? 

8.  Would  the  leaves  in  this  book  cover  all  the  glass  in  the 
school-room  windows  ? 

9.  How  many  square  feet  of  blackboard  in  your  school- 
room? What  did  it  cost?  (Reckon  natural  slate  @  25(*  per 
sq.  ft. ;  artificial  slate  at  1  If  per  sq.  ft. ) 

10.  Draw  on  blackboard  a  diagram  of  the  school-room 
floor,  scale  one  inch  to  the  foot. 

11.  Draw  the  same  on  paper,  scale  one  quarter-ioch  to  the 
foot. 

12.  Draw  a  diagram  of  the  north  wall  using  same  scale. 

13.  Calculate  separately  the  number  of  feet  of  wainscot- 
ing, blackboard,  plaster,  doors,  windows. 

14.  How  many  square  feet  in  a  base-ball  diamond? 

Note.  The  teacher  may  increa.se  this  list  indefiuitely.  The  best  prob- 
lems are  those  drawn  from  the  pupil's  surroundings  or  from  his  studies. 

15.  How  many  square  feet  are  there  in  a  lot  99  feet  wide 
and  208  feet  deep? 

16.  What  is  the  value  of  the  above  lot  at  9i  cents  a  square 
foot? 

17.  If  flooring  costs  3  cents  a  square  foot,  what  did  the 
lumber  in  the  floor  of  your  school- room  cost  ? 

18.  If  it  costs  3  cents  a  square  foot  to  lath  and  plaster  a 
^all,  what  was  the  cost  of  plastering  the  ceiling  of  your 
bchool-room  ? 

19.  If  matting  sells  for  4  cents  a  square  foot,  what  will  it 
cost  to  cover  the  floor  of  your  school-room  with  it? 


DIVISION.  45 


SECTIOT^   Y. 

DIVISION. 

93.  Separate  12  crayons  into  groups  of  4  crayons  each. 
Separate  12  crayons  into  4  equal  groups.  In  each  of  these 
processes  the  12  has  been  separated  into  equal  numbers; 
either  of  the  processes  is  called  Division.  In  the  first  pro- 
cess we  are  given  the  number  to  be  separated  and  the  size  of 
the  equal  groups,  —  that  is,  the  12  are  to  be  measured  off 
into  4's.  This  kind  of  division  is  called  Measurement.  In 
the  second  process  we  are  given  the  number  to  be  separated 
and  the  number  of  equal  groups  to  be  made  of  it.  This 
kind  of  division  is  called  Partition. 

94.     DEFINITIONS. 

1.  Division  is  the  process  of  separating  a  number  into 
equal  numbers. 

2.  Measurement  is  the  process  of  separating  a  number  into 
equal  numbers  of  a  given  size. 

3.  The  Dividend  in  Measurement  is  the  number  that  is 
to  be  separated  into  equal  numbers  of  a  given  size. 

4.  The  Divisor  in  Measurement  is  one  of  the  equal  num- 
bers into  which  the  dividend  is  to  be  separated. 

5.  The  Quotient  in  Measurement  is  the  number  of  equal 
numbers  into  which  the  dividend  has  been  separated. 

6.  Partition  is  the  process  of  separating  a  number  into  a 
given  number  of  equal  numbers. 

7.  The  Dividend  in  Partition  is  the  number  to  be  sepa- 
rated into  a  given  number  of  equal  numbers. 

8.  The  Divisor  in  Partition  is  the  number  of  equal  num- 
bers into  which  the  dividend  is  to  be  separated. 


46  XEW  ADVAXCED  ARITHMETIC. 

9.  The  Quotient  in  Partition  is  one  of  the  equal  numbers 
into  which  the  dividend  has  been  separated. 

10.  The  Remainder  in  each  case  is  the  undivided  part  of 
the  dividend. 

95  1.  In  measurement  the  divisor  and  dividend  have  the 
same  unit.  The  quotient,  being  a  number  of  numbers,  is 
abstract.     The  remainder  is  like  the  dividend. 

2.  In  partition  the  dividend  and  quotient  are  alike 
and  the  divisor  is  abstract.  The  remainder  is  like  the 
di\idend. 

93.  "Which  of  the  following  problems  Ulustrate  measure- 
ment and  which  partition?  Name  divisor,  dividend,  and 
quotient  in  each  case  and  show  how  the  definitions  apply. 

PROBLEMS. 

1.  An  80-acre  field  was  divided  into  10-acre  lots.  How 
many  did  it  make  ? 

2.  A  stage  coach  went  6  miles  an  hour ;  how  many  hours 
were  required  to  go  30  miles  ? 

3.  A  school-room  contains  54  seats  arranged  in  6  equal 
rows ;  how  many  seats  are  there  in  each  row? 

4.  At  3  cents  each,  how  many  oranges  can  be  bought  for 

30  cents? 

5    If  5  barrels  of  flour  cost  $20.  what  is  the  price  per 

barrel? 

6.  If  a  school  of  42  pupils  were  divided  into  6  equal 
classes,  how  many  pupils  would  there  be  in  each  class? 

7.  "With  divisor  and  quotient  in  each  of  above  problems, 
make  a  problem  in  multiplication;  with  dividend  and  quo- 
tient, a  problem  in  division,  and  tell  kind. 

97.    There  are  four  signs  of  division.     They  are :  )  ,  -i- ,  : , 


DIVISION.  47 

The  divisor  is  placed  at  the  left  of  the  first  sign,  at  the 
right  of  the  second  and  third,  and  below  the  fourth. 
Illustrations, 
4)12(3.  12-^4  =  3.  12:4  =  3.  ^^  =  3. 

98.  By  reversing  the  order  of  the  multiplication  table  the 
measmement  and  partition  tables  are  formed.  Show  how 
this  is  done  with  6x8  =  48. 

99.      PROBLEMS    IN    MEASUREMENT. 

1.  Use  2  as  a  divisor  and  give  the  quotients  for  all  num- 
bers from  2  to  19. 

Illustration.  In  2  there  is  one  2.  In  3  there  is  one  2  and 
half  of  another,  etc. 

2.  Using  3  as  a  divisor,  do  the  same  with  numbers  from 
3  to  29. 

3.  With  4  from  4  to  39.  8.  With  9  from  9  to  89. 

4.  With  5  from  5  to  49.  9.  AVith  10  from  10  to  99. 

5.  With  6  from  6  to  59.  10.  With  11  from  11  to  109. 

6.  With  7  from  7  to  69.  11.  With  12  from  12  to  119. 

7.  With  8  from  8  to  79.  12.  With  16  from  16  to  144. 

100.       PROBLEMS    IN     PARTITION. 

1,  Use  2  as  a  di\'isor,  and  give  quotients  for  all  numbers 
from  2  to  19. 

Illustration.     One  half  of  2  is  1.     One  half  of  3  is  \\. 

2.  With  3  to  29. 

Continue  these  exercises  through  the  same  numbers  as  in 
division. 

101.     EXAMPLES  FOR   PRACTICE  IN   MEASUREMENT- 
1.    Divide  8648  by  3. 

FORM. 

.3  )  8648 
^8S2f 


48 


NEW  ADVANCED  ARITHMETIC. 


2. 

7463-^4=?                     7. 

3. 

9608  -^  5  =?                   8. 

4. 

21279  4-  6  =?                   9. 

5. 

39852  ^  7  =?                10. 

6. 

463024  -^  8  =?               11. 

)2. 

Explain  the  eleven  precedi 

FORM. 

3)  8048 

Analysis.  There  are  two  3's  in  8  with  a  remainder  of  2.  Since  the  8 
is  thousands,  the  quotient  and  remainder  are  thousands.  2  thousands  =  20 
hundreds.  20  hundreds  and  6  hundreds  are  26  hundreds.  There  are  eight 
3's  in  26,  with  a  remainder  of  2.  Since  the  26  is  hundreds,  the  quotient 
and  remainder  are  hundreds.  2  hundreds  =  20  tens.  20  tens  and  4  tens 
are  24  tens.  There  are  eight  3's  in  24.  Since  the  24  is  tens,  the  quotient 
is  tens.  There  are  two  3's  in  8,  with  a  remainder  of  2.  2  is  §^  of  3.  Hence 
in  8648  there  are  2882|  threes. 

560074-^9  =? 
730620^  10=? 
582763  -^  11  =? 
9806436 -^  12  =? 
2668937 -^  16  =? 


Analysis.  One  third  of  8  thousands  is  2  thousands,  with  a  remainder 
of  2  thou.sands  2  thousands  =  20  hundreds.  20  hundreds  and  6  hundreils 
are  26  hundreds.  One  third  of  26  hundreds  is  8  hundreds,  with  a  remainder 
of  2  hundreds.  2  hundreds  =  20  tens.  20  tens  and  4  tens  are  24  tens. 
One  third  of  24  tens  is  8  tens.  One  third  of  8  is  2,  with  a  remainder  of  2. 
J  of  2  is  |.     Hence  one  third  of  8648  is  2882|. 

103.  In  problems  like  the  preceding,  in  which  the  product 
of  divisor  and  quotient,  the  remainder,  and  the  new  partial 
dividend  are  remembered  and  not  written,  the  work  is  said 
to  be  by  Short  Division.  The  products  are  not  written  be- 
cause they  are  included  in  the  ordinary  multiplication  table. 

Solve  and  analyze  the  following  rapidly  by  both  methods : 

24937068  -Ml  =  ? 
569478370  4-  16  =  ? 
439450682  ^  12  =  ? 
71.3069458  -^  9  =  ? 
864037926  4-  11  =  ? 


1. 

289634  -;-  9  =  ? 

6. 

2. 

5879639  -4-11  =  ? 

7. 

3. 

608579031  -^  12  =  ? 

8. 

4k 

2.50769037  -^  16  =  ? 

9 

5. 

508763018  ^  12  =  ? 

10. 

DIVISICN.  49 

104.    LONG  DIVISION. 

Illustrative  Problem.     98684  ~  i2  =  ? 

FORM. 
Divisor.  Dividend.  Quotient. 

42  )  98684  (  2349 
84 
146 
126 

208 
188 

"404 
378 

2G    Remainder. 

105.      EXPLANATION. 

There  are  two  42's  in  98.  Multiplying  the  divisor  by  the 
first  term  of  the  quotient,  we  find  their  product  to  be  84, 
which  we  write  beneath  the  partial  dividend.  The  remainder 
is  found  to  be  14.  Since  the  98  is  thousands,  the  quotient 
and  remainder  are  thousands.  14  thousands  =  140  hundreds. 
To  this  number  is  added  the  6  hundreds  of  the  dividend. 
140  hundreds  and  6  hundreds  =  146  hundreds. 

In  146  there  are  three  42's.  Performing  the  multiplication, 
to  find  how  much  146  exceeds  three  42's,  the  remainder  is 
found  to  be  20.  Since  the  146  is  hundreds,  the  quotient  and 
remainder  are  hundreds.  20  hundreds  =  200  tens.  260  tens 
+  8  tens  =  208  tens.  In  208  there  are  four  42's,  with  a 
remainder  of  40.  Since  the  208  is  tens,  the  quotient  and 
remainder  are  tens.  40  tens  =  400  ones.  400  ones  +  4  ones 
=  404  ones.  In  404  there  are  nine  42's,  with  a  remainder 
of  26. 

The  26  may  be  left  as  a  remainder,  or  we  may  indicate 
the  part  of  42  that  it  is  and  join  it  to  the  quotient.  It  is  |§ 
of  42.  Hence  in  98684  there  are  2349  42's  and  If  of 
another,  or  simply  2349 1|  42's. 


50  XEW  ADVANCED  ARITHMETIC. 

To  test  the  work  find  the  product  of  the  divisor  and  quo- 
tient, and  to  it  add  the  remainder.  The  result  should  equal 
the  dividend. 

106.  Analyze  the  same  problem  by  partition,  usmg  the 
analysis  previously  given. 

107.      REMAINDERS. 

Note  that  in  many  problems  we  have  an  integral  quotient 
with  a  remainder,  as,  25  -^  4  =  G  with  a  remainder  of  1  ;  or 
we  may  complete  the  division  obtaining  a  fractional  quotient, 
25  -^  4  =  6^.  AVhether  we  shall  say,  25  divided  into  4's  are 
six  4's  with  a  remainder  of  1,  or  25  divided  into  4's  are  six 
and  one  fourth  4's,  depends  upon  the  nature  of  the  problem 
in  which  these  numerical  relations  are  found. 

25  cents  will  buy  how  many  tin  cups  at  4  cents  each? 

25  cents  will  buy  how  many  pounds  of  sugar  at  4  cents 
each? 

In  which  of  these  problems  is  there  an  undivided  remainder? 

Make  five  concrete  problems  in  measurement  involving 
remainders;  five  in  partition. 

Make  five  concrete  problems  in  measurement  involving 
fractional  quotients ;  five  in  partition. 

108.     RULE  FOR  LONG  DIVISION. 

At  the  left  of  the  dividend  take  a  partial  dividend  that 
win  contain  the  divisor,  and  place  the  first  term  of  the 
quotient  at   the  right. 

Multiply  the  divisor  by  the  term  of  the  quotient  thus 
obtained,  u-rite  the  product  beneath  the  partial  dividend, 
and  subtract. 

To  the  remainder  thus  obtained  annejr  the  ne.rt  term  o/ 
the  dividend  for  a  second  partial  dividend,  and  proceed 
as  before. 

Note  1.  If  a  new  partial  dividend  ■will  not  contain  the  divisor,  place  a 
cipher  in  the  quotient,  annex  the  next  term  of  the  dividend,  and  proceed 
as  before. 


DIVISION. 


51. 


NoTs  2.     In  obtaining  any  term  of  the  quotient,  compare  the  first  term 
of  the  divisor  with  the  first  part  of  the  partial  dividend. 

Note  3.     I'rove  the  problems  as  in  short  division. 


109.     EXAMPLES  FOR   PRACTICE. 

Note.  Long  Division  usually  begins  with  13  as  a  divisor.  In  solviug 
the  following  problems  let  the  pupils  make  and  refer  to  the  multiplication 
tables  for  13,  14,  etc.,  to  19,  until  they  can  find  readily  any  term  of  the 
quotient.     Thus?  2  X  13  —  26;  3  X  13  =:  39,  etc. 

I.  Divide  by  13:  31668;  56641;  49784;  73433;  987205; 
1266238. 

2o  Divide  by  14  3430;  4536;  6490;  80132;  96039; 
1219b53;  1285310. 

3.  Divide  by  15:  3705;  5777;  10269;  91185;  59947; 
131549. 

4.  Divide  by  16:  3952;  97264;  140318;  942448; 
1376048;  892751;  394887;  1427759;  8895779;  1086331. 

5.  Divide  by  17:  228225;  453151;  821993;  998746; 
1201847;  15943688;  1482275;  6293332;  1172540. 

6.  Divide  by  18:  244153;  438753;  644508;  988513; 
877858;  1313982;  1743954;  1507995. 

7.  Divide  by  19:  6574;  32908;  268669;  586915  •, 
1290105;  1078000;  1798712;  1515876;  1311040; 
1410003. 

8.  Divide  68324  by  20;  30;  40;  50;  60 

9.  Divide  47906  by  21 ;  31 ;  41 ;  51 ;  61 

10.  Divide  74583  by  22  ;  32  ;  42  ;  52 

II.  Divide  194873  by  23;  33;  43;  53 

12.  Divide  108460  by  24  ;  34  ;  44  ;  54 

13.  Divide  298765  by  25  ;  35  ;  45  ;  55 
14  Divide  370625  by  26  ;  36  ;  46  ;  56 

15.  Divide  872056  by  27;  37;  47;  57 

16.  Divide  476921  by  28;  38;  48;  58 


60; 

70; 

80; 

90, 

61; 

71; 

81; 

91. 

62 

;    72; 

82; 

92. 

63 

;   73; 

83; 

93. 

64 

;   7-^ 

84; 

94. 

65 

;   75; 

85; 

95. 

66 

;   76; 

86; 

96. 

67 

;   77; 

87; 

97. 

68 

•   78; 

88; 

98. 

52  NEW  ADVANCED  ARITHMETIC. 

17.  Divide  572183  by  29;  39;  49;  59;  69;  79;  89;  99. 

18.  109278^234;    109278 -^  467o 

19.  934605-^567;    934605-^1667. 

20.  27732494  -^  4658 ;  27732494  -^  5943. 

21.  794006387  h-  568946. 

22.  74320876  -i-  6958. 

23.  14173345  -f-  2005. 

24.  14173345  -^  7069. 

25.  602305812  -^  70003. 

26.  602305812  -^  8604. 

27.  A  bought  37  acres  of  land,  for  which  he  paid  S2,664. 
"What  was  the  price  per  acre? 

Measurement  or  partition?     Why? 

Analysis.  Since  37  acres  cost  $2,664,  each  acre  cost  one  thirty-seventh 
of  $2,664. 

28.  At  $72  an  acre,  how  many  acres  of  land  can  b3 
bought  for  S2,664? 

Measurement  or  partition  ?     Why  ? 

Analysis.  Since  each  acre  cost  $72,  as  many  acres  can  be  bought  for 
$2,664  as  there  are  72's  in  2,664.  There  are  thirty -seven  72's  in  2,664. 
Hence,  for  $2,664,  37  acres  can  be  bought,  at  $72  an  acre. 

Analyze  each  of  the  following : 

29.  At  43  miles  an  hour,  how  long  will  it  take  a  train  to 
run  1,677  miles? 

30.  If  a  train  run  1,677  miles  in  39  hours,  what  is  the  rate 
per  hour? 

31.  38  pieces  of  cloth  cost  ^1,786.  What  was  the  average 
price  ? 

32  At  $47  a  piece,  how  many  pieces  of  cloth  can  be 
bought  for  $1,786? 

33.  A  farmer  sold  his  wheat  at  97  cents  a  bushel,  receiv- 
ing $353.08.     How  many  bushels  did  he  sell? 


DIVISION.  53 

34.  A  farmer  sold  364  bushels   of   wheat   for   $353.08. 
What  was  the  price  per  bushel? 

35.  15  bu.  3  pk.  4  qt.  of  oats  were  divided  into  4  equal 
[Ales.     What  amount  was  there  in  each  pile? 

36.  Divide  24  gal.  3  qt.  1  pt.  2  gi.  of  vinegar  into  7  equal 
parts.     What  is  the  amount  in  each  part? 

37.  The  divisor  is  328,  the  quotient  407,  and  the  remain- 
der 279.     What  is  the  dividend? 

38.  Make  a  rule  for  finding  the  dividend  when  the  divisor, 
quotient,  and  remainder  are  given. 

39.  The  dividend  is  364,280,  the  quotient  877,   and  the 
remainder  325.     What  is  the  divisor? 

40.  Make  a  rule  for  finding  the  divisor  when  the  dividend, 
remainder,  and  quotient  are  given. 

110.    DiTision  by  10,  100,  etc. 
1.   8640 -MO.        2.    4900-^100.         3.    596000 -MOOD. 

4.  Make  a  rule  based  on  the  three  preceding  problems. 

5.  58764  -^  100. 

What  is  the  quotient  in  Problem  5?  the  remainder?    Make 
a  rule  for  such  cases.     Does  this  include  every  case? 

RULE  FOR  DIVIDING  BY  A  POWER  OF  10. 
Cut  off  fyoui  the  right  of  the  iliviaend  as  many  fignres 
as  there  are   ciphers  in  the   dirisor.     The   part  thus   cut 
off  is  the  remainder,  and.  the  rest  of  the  dii'idend  is  the 
Qtiotient. 

6.  Divide  79640  by  10;   100;   1000^   10000. 

7.  Make  and  solve  six  problems  Hire  the  above. 

111.    Changing  problems  from   measurement  to  partition, 
or  from  partition  to  measurement, 

1.    Paid  75  cents  for  oranges  at  5  cents  each.     How  many 
did  1  buy? 

5A 


54  NEW  ADVANCED  ARITHMETIC. 

Give  analysis  like  that  of  Problem  28,  Art.  109. 
Change  to  partition. 

Analysis.  If  the  oranges  had  cost  cue  cent  each,  75  cents  would  have 
bou"-ht  75  oranges.  Since  they  cost  5  cents  each,  75  cents  bought  one  fifth 
oi  75  oranges.     One  fifth  of  75  oranges  is  15  oranges. 

2.  25  acres  of  land  cost  $6.30.  What  was  the  average 
price  per  acre? 

Give  analysis  like  that  of  Problem  27,  Art.  109. 
Change  to  measurement. 

Analysis.  If  each  acre  had  cost  $1,  25  acres  would  have  cost  $25. 
Since  25  acres  cost  $650,  each  acre  cost  as  many  times  |l  as  there  are  25's 
a  650.     There  are  26  25's  in  650.     Hence,  each  acre  cost  $26. 

3.  Review  Problems  29-34,  inclusive.  Art.  109,  making 
the  changes  as  indicated  above. 

112.    Division  by  Aliquot  Parts. 

Review  the  aliquot  parts  of  100  and  1000  given  in  multi- 
plication. 

FORM. 

73800 


1.  Divide  73800  by  37^.  738 

5904 

1968 

Analysis.     37iis|oflOO. 

If  the  divisor  were  100,  the  quotient  would  be  7.38. 

If  the  divisor  were  \  of  100,  the  quotient  would  he  eiglit  738's,  which 
are  5904. 

Since  the  divLsor  is  |  of  100,  the  (luotient  is  i  of  5904,  which  is  1968. 

2.  Divide  465500  by  25 ;    37^;    62^;    87i. 

3.  Divide  39683000  by  125 ;    375;    625;    875. 

4.  Divide  269500  by  8JL  ;    16^;    33^;    U?,]    58 J., ;    91§. 

5.  Divide  5005000  by  162  ;    331  ;    41f,;    58';    91|. 


Dl  VISION.  55 

6.  How  many  dozens  of  eggs,  at  12^  cents  a  dozen,  can 
be  bought  for  $2  ?  $5?  $3.75?  $6.50? 

Analysks.  (a)  If  the  egi;s  were  $1  a  dozeu,  $2  would  bu\-  2  dozens. 
Since  the  eggs  are  ^  of  $1  a  dozen,  $2  will  buy  8X2  dozens,  which  ~  16 
dozeus. 

Or  (b)  At  12.^  cents  a  dozen,  $1  will  buy  8  dozens,  and  $2  will  buy  twice 
8  dozens,  which  equals  16  dozens. 

7.  How  many  pounds  of  butter  can  be  bought  for  $7.50 
at  25  cents  a  pound?  at  33^  cents?  at  16|  ceuts?  at  50 
cents  ? 

8.  How  many  yards  of  cloth  can  be  bought  for  $10.50  at 
37 J  cents  a  yard?  at  <d2^  cents?  at  87 i  cents? 

113.    Division  by  Factors. 

Division  may  sometimes  be  simplified  by  dividing  by  the 
factors  of  the  divisor,  thus  performing  the  problem  by 
"_ short  division  "  instead  of  by  "  long  division."  The  only 
difficulty  is  in  understanding  the  remainders. 

1.  Divide  6936  by  12. 

Explanation.  ^^  of  6936  =  ^  of  i  of  6936.  \  of  6936  = 
1734.  I  of  1734  =  578. 

2.  11745-^15  =  ?         4.  150010 -=- 35  =  ? 

3.  135072  ^  24  =  ?        5.  5377792  -^  56  =  ? 

114.  To  Find  the  True  Remainder. 

8661  ^  42  =  ? 

Explanation.  42  =  2  X  3  X  7.  This  problem,  and  all 
similar  problems,  may  be  read  in  this  way :  How  many  42's 
are  there  in  8661 .'' 

There  are  4330  2's  in  8661,  and  a  remainder  of  1.  If 
4330  2's  be  separated  into  groups,  each  of  which  shall  con- 
tain three  2's,  there  will  be  1443  such  groups,  with  a  re- 
mainder of  one  2.  Each  of  these  groups  contains  three  2's, 
which  equals  one  6. 


56  NEW  ADVANCED  ARITHMETIC. 

Separating  1443  6's  into  groups  each  of  which  shall  con- 
tain seven  6's,  we  find  that  there  are  206  such  groups,  and 
a  remainder  of  one  6.  8661  contains  206  42's,  with  a  re- 
mainder of  one  1,  one  2,  and  one  6,  whose  sum  is  9. 

The  first  remainder  in  such  cases  is  ones;  each  of  the 
units  in  the  second  remainder  equals  the  first  divisor;  each 
of  the  units  in  the  third  remainder  is  obviously  equal  to  the 
product  of  the  first  and  second  divisor,  and  so  on. 

Explain  also  as  a  problem  iu  partition. 

115.     RULE  FOR   FINDING  THE  TRUE  REMAINDER. 

Multiply  the  remainder  obtained  by  each  dirision  by 
all  of  the  preceding  tlivisors.  To  the  sum  of  these  products 
add  the  first  remainder. 

1.  43259  ^  110.      110  =  2  X  5  X  11. 

2.  64727  -^  3«5.      385  =  5  X  7  X  11. 

3.  583077^308.     308=4x7x11. 

4.  9386457  ^  343. 

116.  1.  $10.72^4  =  ?  (Explain  by  partition.)  $8.76 
-^  6  =  ?     $78.52  ^  8  =  ? 

2.  10.72  -f-  4  =  ?     8.76  -^  6  =  ?     78.52  ^  8  =  ? 

3.  87.437 -^  7;  9.6864^16;   .624^25;   .0834-^75. 

4.  .00384^4;  by  48;  by  64. 


.16842 


24;  by  225;  by  192. 


117.     OUTLINE  REVIEW. 


1.  Numbers  may  be  : 

(1)  Expressed. 

(2)  United. 

(3)  Separated. 

2.  Numbers  are  expressed  by  : 

(1)  The  Arabic  method. 

(2)  The  Roman  method. 


DI  VISION.  57 

3.  Numbers  are  united  by  : 

(1)  Addition. 

(2)  Multiplication. 

4.  In  addition  the  numbers  are  alike,  and  may  be  equal 
or  unequal. 

5.  In  multiplication  the  numbers  are  alike  and  equal. 

6.  The  product  is  of  the  same  unit  as  the  multiplicand. 
The  multiplier  is  abstract. 

7.  Numbers  may  be  separated  by : 

(1)  Subtraction. 

(2)  Measurement. 

(3)  Partition. 

8.  In  subtraction  a  number  is  separated  into  two  parts 
that  may  be  equal  or  unequal. 

9.  The  subtrahend  and  remainder  are  of  the  same  unit  as 
the  minuend. 

10.  In  measurement  and  partition  numbers  are  separated 
into  two  or  more  equal  parts.  In  measurement  the  size  of 
the  parts  is  given,  and  the  number  is  required. 

11.  The  dividend  and  divisor  are  of  the  same  denomina- 
tion, and  the  quotient  is  abstract 

12.  In  partition  the  number  of  parts  is  given,  and  the 
size  is  required. 

13.  The  dividend  and  quotient  are  of  the  same  unit,  and 
the  divisor  is  abstract. 

118.      ORAL    EXERCISES. 

1.  4  times  6  are  how  many  times  8  ? 

2.  6  times  8  are  how  many  times  12? 

3.  4  times  14  are  how  many  times  8? 

4.  5  times  12  are  how  many  times  15? 

5.  8  times  9  are  how  many  times  4? 

6.  9  times  12  are  how  many  times  6  times  3? 

7.  12  times  8  are  how  many  times  4  times  6? 


58  NEW  ADVANCED  ARITHMETIC. 

8     o  times  6  plus  4  times  8  are  how  many  times  2? 

9.  3  times  IG  plus  4  times  6  are  how  many  times  2  times 
9? 

10.  4  times  16  less  3  times  4  are  how  many  times  13? 

11.  5  times  13  plus  2  times  5  are  how  many  times  25? 

12.  4  times  17  plus  3  times  9  are  how  many  times  19? 

13.  4  times  19  plus  2  times  7  less  3  times  17  are  how 
many  times  13? 

14.  5  times  16  are  how  many  times  10? 

15.  4  times  16  plus  6  are  how  many  times  7? 

16.  If  2  oranges  cost  10  cents,  what  will  7  oranges  cost? 

17.  If  9  yards  of  muslin  cost  108  cents,  what  will  7  yards 
cost? 

18.  If  11  books  cost  $4.40,  what  will  6  books  cost? 

19.  If  a  man  travels  72  miles  in  4  hours,  how  many  miles 
will  he  travel  in  5  hours  going  at  the  same  rate? 

20.  If  12  acres  of  oats  yield  480  bushels,  how  many 
bushels  will   17  acres  yield  at  the  same  average? 

21.  If  $24  buy  12  yards  of  cloth,  how  many  yards  will  $36 
buy  at  the  same  price? 

22.  If  11  cords  of  wood  cost  $44,  how  many  cords  can  be 
bought  for  $72  at  the  same  price?  for  $84?  for  $92?  for 
$68? 

23.  If  7  men  can  do  a  piece  of  work  in  10  days,  in  how 
many  days  can  2  men  working  at  the  same  rate  do  the  work? 
5  men?  14  men? 

24.  If  8  men  can  do  a  piece  of  work  in  12  days,  how 
many  men  would  be  required  to  do  the  same  work  in  6  days? 
in  8  days?  in  16  days?  in  24  days? 

25.  Sold  3  dozen  eggs  at  12  cents,  and  2  pounds  of  butter 
at  24  cents.     What  is  the  change  out  of  a  dollar? 

26.  Sold  5  articles  at  10  cents,  2  articles  at  12  cents,  and 
2  at  10  cents.     Find  change  out  of  a  dollar. 


DIVISION.  yy 

27.  Sold  3  articles  at  15  ceuts,  3  at  10  ceuts,  and  1  at  8 
cents.     Find  change  for  a  dollar. 

28.  Sold  4  articles  at  15  cents,  2  at  8  cents,  and  1  at  14 
cents.     Find  change  for  a  dollar. 

29.  In  how  many  days  can  5  men  earn  as  much  as   10 
men  can  earn  in  4  days? 

30.  Five  9's  are  how  many  15's? 

31.  How  many  days  will  be  required  for  8  men  to  do  the 
work  performed  by  12  men  in  4  days? 

32.  Twelve  7's  are  how  many  2rs? 

33.  If  4  oranges  are  worth  12  apples,  how  many  apples 
are  23  oranges  worth? 

34.  How  many  9's  in  three  2rs  +  18? 

35.  What  is  the  cost  of  24  bushels  of  apples  if  16  bushels 
cost  S9. 60? 

36.  7  times  8,  —  5  is  how  many  17's? 

37.  In  12  days  8  men  will  earn  as  much  as  how  many  in 
16  days? 

38.  16  times  37i  is  how  many  times  12^? 

39.  How  many  bushels  of  oats  at  25  cents  can  be  bought 
for  125  bushels  of  wheat  at  75  cents? 

40.  25  times  16|  is  how  many  times  83i? 

41.  Bought  13  barrels  of  flour  for  $91;  gave  9  of  them 
for  apples  at  $3  a  barrel.     How  many  were  received? 

42.  8  times  12  —  9  are  how  many  29's? 

43.  Bought  13  barrels  of  flour  for  §78;  gave  6  of  them 
for  12  barrels  of  apples.     What  were  the  apples  a  barrel? 

119.     THE   LAW    OF    SIGNS. 

The  brackets  [],  parentheses  (),  braces  {},  and  ^-incu- 
lum  are  called  symbols  of  aggTegation  ;  enclosed  expres- 

sions are  called  bracketed  expressions;    operations  within 


60  NEW  ADVANCED  ARITHMETIC. 

the  bracketed  expressions  must  be  performed   first,   thus: 
12  -  [5  X  2]  +  [30  -^  6]  =  12  -  10  +  5  =  7. 

If  no  brackets  occur,  multiplications  and  divisions  are 
performed  in  order  before  any  additions  or  subtractious, 
thus :   12  —  5  X  2  +  30  -^  5  X  3  =  12  —  10  +  18  =  20. 

EXERCISES. 


1.  (8-2  +  3)  X  (6  +  7-  9)  =? 

2.  4  X  [6  -  {11  -  (5  +  3  }  +  2]  =? 

3.  19  +  3x3-64-^8  +  5x4=? 

4.  [30  -^  5  X  2  +  9  X  2]  ^  [10  +  60  ^  12]  =? 

5.  [60  -^  6  +  4  X  3]  ^  [15  -  4]  X  [90  X  2  -f-  3  +  40]  =? 

120.     1.    (86429  -  4786)  -  (7512  -  482). 

2.  86429  -  4786  -  7512  -  482. 

3.  The  divisor  is  879,  the  quotient  46,  and  the  remainder 
23;  what  is  the  dividend? 

4.  The  remainder  is  279  and  the  subtrahend  673 ;  what  is 
the  minuend? 

5.  The  product  is  4212,  and  the  multiplier  78 ;  what  is  tlie 
multiplicand  ? 

6.  The  minuend  is  964,  and  the  remainder  278;  what  is 
the  subtrahend? 

7.  The  product  is  195  and  the  multiplicand  39  ;  what  is 
the  multiplier? 

8.  Find  the  sum  of    3  bu.  2  pk.  4  qt. 

5  "  1  "  6  " 

7  "  3  "  2  " 

4  "  0  "  7  " 

12  "  3  "  5  " 


9.    A  merchant, having  23  gal.  2  qt.  1  pt.  2  gi.  of  vinegar, 
sold  17  gal.  3  qt.  1  pt.  3  gi.     What  did  he  have  left? 


DIVISION.  61 

10.  A  had  125  bushels  of  corn;  B  had  trrice  as  much  and 
25  bushels  more ;  C  had  as  much  as  both  A  and  B ;  D  had  us 
much  as  the  difference  between  A's  and  B"s.  The  corn  was 
sold  at  37^  cents  a  bushel.     What  did  it  bring? 

11.  How  many  seconds  are  there  in  a  minute?  minutes 
in  an  hour  ?   hours  in  a  da}^  ?   days  in  a  week  ? 

12.  A  man  worked  6  days  of  8  h.  23  m.  46  s.  each.  How 
many  hours,  minutes,  and  seconds  did  he  work? 

13.  In  6  days  a  man  put  in  50  h.  22  m.  36  s.  What  was 
his  daily  average? 

14.  (24  -  6)  -^  3  =  ?     24  -  6  H-  3  =  ? 

15.  (3  +  4)  X  2  =  ?     3  +  4x2  =  ? 

16.  8  X  2  —  (6  —  2)  H-  2  X  4  +  2  X  13  -i-  5  =  ? 

17.  I  bought  125  acres  of  land  at  '^^0  an  acre.  I  spent 
81,250  to  fence  it.  At  what  price  an  acxe  must  I  sell  it  to 
gain  01,000? 

18.  If  12  men  can  do  a  piece  of  work  in  15  days,  in  how 
many  days  can  20  men  do  it? 

19.  18  men  undertook  a  piece  of  work  that  would  take 
them  24  days;  when  it  was  half  done,  10  of  them  left.  In 
how  many  days  could  those  remaining  finish  it? 

20.  A  farmer  bought  20  pounds  of  sugar  at  6  cents  a 
pound;  10  yards  of  cloth  at  75  cents  a  yard;  2  pounds  of 
tea  at  50  cents  a  pound;  15  pounds  of  coffee  at  20  cents  a 
pound.     He  sold  the  merchant  an  equal  number  of  bushels 

I  potatoes  and  apples,  getting  40  cents  a  bushel  for  the 
former,  and  60  cents  a  bushel  for  the  latter,  and  received  70 
cents  in  change.     How  many  bushels  of  each  did  he  sell? 

21.  A  merchant  bought  48  yards  of  cloth  at  62  cents  a 
yard,  and  81  yards  at  75  cents  a  yard.  He  sold  the  former 
at  81  cents  a  yard,  and  the  latter  at  such  a  price  as  to  gain 
820.37  on  the  whole  transaction.  At  what  price  did  he  sell 
it  per  yard? 


62  ■  NEW  ADVANCED  ARITHMETIC. 

22.  How  many  States  equal  in  area  to  New  England  can 
be  made  from  Texas?  What  is  the  remainder?  It  is  nearest 
the  area  of  what  State?     Is  about  what  part  of  it? 

23.  The  total  area  of  the  cotton  States  is  how  many  times 
that  of  New  England? 

24.  Compare  that  portion  of  the  United  States  lying  west 
of  the  Mississippi  River  with  the  portion  lying  east  of  it. 

25.  The  populations  of  the  New  England  States  in  1890 
were  as  follows:  Me.,  666,086;  N.  H.,  376,530;  Vt., 
349.290;  Mass.,  2,238,943  ;  R.  I.,  345,.-)06  ;  Conn.,  746,258. 
Find  the  population  per  square  mile  for  each.  Arrange  them 
in  the  order  of  the  density  of  population. 

26.  If  your  State  were  as  densely  populated  as  Massachu- 
setts, how  many  people  would  it  contain  ? 

27.  Draw  a  row  of  6  square  inches.  How  many  such 
rows  are  needed  to  make  24  square  inches?  36  square 
inches?     60?     144? 

28.  Your  tablet  page  is  9  inches  long.  If  it  were  one  inch 
wide,  what  would  be  its  area?  How  many  inches  wide  must 
it  be  to  contain  45  square  inches?  63  square  inches?  81? 
108?    144? 

29.  The  sheet  upon  which  I  write  contains  80  square 
inches ;  it  is  10  inches  long ;  how  wide  is  it? 

Analysis.  Were  the  sheet  one  inch  wide,  it  would  contain  10  siiuare 
inches.  To  contain  80  square  inches  it  must  be  .as  many  inches  wide  as 
there  are  lO's  in  80.  There  are  8  lO's  in  80.  Hence  the  sheet  is  8  inches 
wide. 

30.  In  the  window  at  ni}-  elbow,  each  pane  is  14  inches 
wide  and  contains  392  square  inches.     How  long  is  each? 

31.  What  is  the  width  of  the  desk-top  upon  which  I  write? 
Its  area  is  999  square  inches,  and  its  length  37  inches. 

32.  The  floor  of  this  room  is  26  feet  wide  and  contains  819 
square  feet.  How  long  is  it?  How  many  square  yards  of 
linoleum  must  I  buy  to  cover  it? 


DIVISION.    ■  G3 

33.  A  40-acre  field  is  1,320  feet  square ;  how  wide  a  strip 
along  one  side  contains  an  acre  (43,560  square  feet)?  How 
many  corn-rows,  3  feet  8  incites  apart,  can  be  planted  in  this 
strip  the  long  way? 

34.  A  man  bought  a  piece  of  land  40  rods  long  and  24 
rods  wide.  He  bought  a  second  piece  containing  as  many 
square  rods  but  30  rods  wide  ;  what  was  its  length?  He  paid 
$72.50  an  acre  (160  square  rods)  for  the  two  pieces ;  what 
did  they  cost? 

35.  A  town  lot  having  a  depth  of  161  feet  and  containing 
19,642  square  feet  sold  for  §2,013;  what  was  the  price  per 
front  foot? 

36.  Lumber  is  sold  by  the  thousand  feet,  the  unit  being  a 
board  12  inches  long.  12  inches  wide,  and  one  inch  thick. 
If  flooring  is  worth  $25  a  thousand,  what  did  the  lumber  cost 
to  floor  your  school-room? 

37.  If  at  the  above  price  the  flooring  for  a  room  cost  832, 
how  many  square  feet  were  there  in  the  floor?  If  the  room 
was  40  feet  long,  what  was  its  width? 

38.  A  man's  crop  of  oats  brought  him  $493.90,  the  selling 
price  being  22  cents  a  bushel.  The  field  in  which  he  raised 
them  was  80  rods  wide  and  90  rods  long.  What  was  the 
average  jneld  per  acre? 

39.  Bought  wheat  at  63  cents  a  bushel,  paying  for  it 
$2,844.12;   how  many  bushels  were  bought? 

40.  At  21  cents  a  bushel,  how  many  bushels  will  8188.74 
buy? 

41.  The  President  of  the  United  States  receives  $50,000 
a  year;  how  much  is  that  for  each  day  of  1896? 

42.  An  Illinois  farmer  sold  his  farm,  consisting  of  320 
acres,  at  882.50  an  acre,  and  invested  the  proceeds  in  west- 
ern land  at  $30  an  acre ;  how  many  acres  did  he  buy? 

43.  Sold  a  piece  of  property  for  $3,825;  anotiier  for 
$4,682  ;  a  third  for  $5,620.     After  paying  a  debt  of  $6,327 


64  T^EW  ADVANCED  ARITHMETIC, 

he  invested   the  remainder  in  land  at  $65  an   acre.      Ho-w 
many  acres  did  he  get? 

44.  How  much  will  it  cost  to  cover  your  school-room  floor 
with  mattii]g  costing  32  cents  a  yard? 

45.  If  it  cost  833.60  to  cover  the  floor  of  a  room  36  feet 
long  with  matting  a  yard  wide,  worth  28  cents  a  yard,  what 
is  the  width  of  the  room? 

46.  Place  4  inch-cubes  in  line.  Place  3  such  rows  side  by 
side.  Place  another  cube  upon  each  of  the  12  cubes.  How 
long  is  the  solid  you  have  formed?  How  wide?  How  high? 
How  many  cubic  inches  in  a  solid  4  in.  x  3  in.  X  2  in.? 

47.  How  many  inch-cubes  are  needed  to  build  a  solid  6  in. 
X  5  in.  X  4  in.? 

48.  How  many  inch-cubes  will  fill  a  chalk  box  6  in.  X  3 
in.  X  3  in.  ? 

Note.  The  number  of  cubic  units  that  a  box  or  vessel  holds  is  called 
its  capacity. 

49.  What  is  the  capacity  of  an  upper  drawer  of  my  desk 
which  is  21  in.  X  10  in.  X  3  in.  ? 

Analysis.  A  drawer  1  inch  lone;,  1  inch  wide,  and  I  inch  deep  holds  1 
cubic  inch.  A  drawer  21  inches  long,  1  inch  wide,  and  1  incli  deep  holds 
21  times  1  cubic  inch,  which  is  21  cubic  inches.  A  drawer  21  inches  long, 
10  inches  wide,  and  1  incli  deep  Iiolds  10  times  21  cubic  inches,  wliich  is  210 
cubic  inches.  A  drawer  21  inches  long,  10  inches  wide,  and  3  inches  deep 
holds  3  times  210  cubic  inches,  which  is  630  cubic  inches. 

50.  The  next  drawer  is  21  in.  X  10  in.  X  4  in.  What  is  its 
capacity? 

51.  The  lowest  drawer  contains  1,6^0  cubic  inches.  Its 
bottom  is  21  in.  X  10  in.  How  many  cubic  inches  are  needed 
to  cover  the  bottom  1  inch  deep?  How  many  such  la^-ers  to 
fill  the  drawer?     What  is  the  depth  of  the  drawer? 

52.  231  cubic  inches  equal  1  gallon.  A  wagon  tank  is  10 
feet  long  and  33  inches  wide,  inside  measure.  3G0  gallons 
fill  it  to  what  depth? 


DIVISION.  65 

53.  "What  is  the  capacity  of  my  school-room,  30  ft.  X  24 
ft.  X  15  ft.? 

54.  A  cubic  foot  of  air,  at  70  degrees  Fahrenheit,  weighs 
525  grains.     How  many  grains  of  air  in  the  school-room? 

55.  7000  grains  equal  1  pound.  How  many  pounds  of 
air  in  your  school-room? 

56.  The  coined  gold  of  the  world  is  estimated  at  10,800 
cubic  feet.  To  what  height  will  it  reach  if  piled  uniformly 
so  as  to  cover  your  school-room  floor? 

Note.     The  number  of  cubic  units  iu  a  solid  is  called  its  volume. 

57.  What  is  the  volume  of  a  brick  8  hi.  X  4  in.  X  2  in.? 

58.  What  is  the  volume  of  a  pine  sill  S  in.  X  12  in.  X 
24  ft.  ? 

Note.  In  describinf^  timbers  tlie  dimensions  arc  given  in  this  order: 
thickness,  widtii,  length. 

59.  Dry  pine  weighs  29  pounds  to  the  cubic  foot.  What 
is  the  weight  of  the  sill? 

60.  How  many  men  are  needed  to  carry  it,  no  man  lifting 
more  than  200  pouiids? 

61.  Joliet  limestone  weighs  1 60  pounds  to  the  cubic  foot. 
What  is  the  weight  of  a  stone  step  6  in.  x  o  ft.  X  12  ft.  ? 

121.    PROPERTIES    OF    NUMBERS. 

1.  Numbers  are  Integral,  Fractional,  or  Mixed. 

2.  An  Integral  Number  is  a  number  of  whole  units. 

3.  Integral  numbers  are  classified  with  respect  to  their 
divisibility  into  Composite,  Prime,  Even,  and  Odd. 

4.  A  Factor  of  a  number  is  one  of  the  two  or  more  inte- 
gral numbers  which  being  multiplied  together  will  produce 
that  number.     It  is  consequently  a  divisor  of  that  number, 

5.  A  Composite  Number  is  a  number  that  has  factors 
beside  itself  and  1. 

4,  10,  35,  are  composite  numbers. 


66  NEW  ADVANCED  ARITHMETIC. 

6.  A  Prime  Number  is  a  number  that  has  no  factors 
except  itself  and   1. 

1,  2,  3,  5,  etc.,  are  prime  numbers. 

7.  Two  numbers  are  prime  to  each  other  when  they  have 
no  common  factor. 

8.  An  Even  Number  is  a  number  that  contains  2  as  a 
factor. 

4,  6,  8,  10,  etc.,  are  even  numbers. 

9.  An  Odd  Number  is  a  number  that  does  not  contain  2 
as  a  factor. 

3,  5,  7,  y,  etc.,  are  odd  numbers. 

Note.  Readiuess  in  factoring  depends  upon  knowing  certain  proper- 
ties of  numbers,  beuce  they  should  be  carefuUy  noted. 

122.    PRINCIPLES. 

1.  A  factor  of  a  number  is  a  factor  of  any  number  of 
times  that  number. 

This  is  obvious,  since  every  time  the  number  is  repeated 
the  factor  is  repeated. 

Illastration.  5  is  a  factor  of  10.  It  must  then  be  a  fac- 
tor of  any  number  of  lO's. 

2.  A  common  factor  of  two  numbers  is  a  factor  of  their 
sum. 

Illustration.  3  is  a  factor  of  6  and  of  9.  It  must  then 
be  a  factor  of  15.  For  6  is  two  3's  and  9  is  three  3's,  hence 
the  sum  of  6  and  9  is  five  3's. 

Demonstration.  Since  encli  number  is  some  number  of 
times  tlie  common  factor,  their  sum  must  be  some  number 
of  times  the  common  factor. 

3     A  divisor  of  tw^o  numbers  is  a  divisor  of  their  difference. 

Illustration .     If  o  is  a  connnon  factor  of  two  numbers,  as 

15  and  25,  each  must  be  composed  of  5's.     If  one  exceeds 


DIVISION.  67 

the  other  it  must  be  because  it  has  more  5's,  hence  the  dif- 
ference must  be  »"?. 

Demonstration.  Since  each  is  some  number  of  times  the 
common  divisor,  if  they  differ  it  is  because  one  of  them  con- 
tains the  common  divisor  more  times  than  the  other,  hence 
their  difference  is  divisible  by  the  common  divisor. 

123.    TESTS    OF    DIVISIBILITY. 

1.  Any  number  is  divisible  by  2  if  its  right-hand  figure  is 
2,  4,  6,  8,  or  0. 

2.  Any  number  is  divisible  by  3  if  the  sum  of  the  ones 
expressed  by  its  digits  ia  divisible  by  3. 

Thus,  263457  is  divisible  by  o  because  the  sum  of  2,  G,  3, 
4,  5,  and  7  is  divisible  by  3. 

3.  A  number  is  divisible  by  4  if  the  number  expressed 
by  the  two  right-hand  digits  is  divisible  by  4. 

Thus,  39484  is  divisible  by  4  because  84  is  divisible  by  4. 

4.  A  number  is  divisible  by  5  if  the  right-hand  digit  is 
0  or  5. 

5.  Any  number  is  divisible  by  6  if  divisible  by  2  and  3. 

Thus,  27684  is  divisible  by  6  because  it  is  even,  and  the 
cum  of  the  ones  expressed  by  its  digits  is  divisible  by  3. 

6.  A  number  is  divisible  by  8  if  the  number  expressed 
by  the  three  right-hand  digits  is  divisible  by  8. 

Thus,  6252144  is  divisible  by  8  because  144  is  divisible 
by  8. 

7.  A  number  is  divisible  by  9  if  the  sum  of  the  ones 
expressed  by  its  digits  is  divisible  by  9. 

Thus,  8645373  is  divisible  by  9  because  8  +  6  +  4  +  5-f3 
+  7  -I-  3  =  36,  a  multiple  of  9. 

8.  A  number  is  divisible  by  10  if  the  right-hand  figure 
is  0. 


68  NEW  ADVANCED   ARITHMETIC. 

9.  A  number  is  divisible  by  11  if  the  sum    of  the  ones 

expressed  by  the  digits  iii  the  odd  orders  equals  the  sum 
of  the  ones  expressed  by  the  digits  in  t^  ,  even  orders,  or 
if  the  difference  of  these  sums  is  a  multiple  of  11. 

Esamiue  8730645. 

The  odd  orders  are  occupied  by  5,  6,  3,  and  8.  The  sum 
of  these  numbers  is  22.  The  even  orders  are  occupied  by 
4,  0,  7.  The  sum  of  these  numbers  is  11.  22  —  11  =  11; 
hence,  this  number  is  divisible  by  11. 

10.  The  tests  given  are  those  most  commonly  used. 
Others  might  be  added,  but  they  are  of  little  practical 
value. 

Make  a  test  for  12,  for  15,  and  for  18. 

11.  When  shall  we  conclude  that  a  number  is  prime? 

Examine  the  number  293.  Is  it  divisible  by  2?  Why? 
By  3?  Why?  Should  4  be  tried?  Why?  Is  it  divisible 
by  5?  by  7?  by  11?  by  13?  by  17?  Why  not  try  6,  8,  9, 
10,  12,  14,  and  16?  What  numbers  have  really  been  tried 
as  divisors?  Examine  the  quotients  obtained  by  these  suc- 
CL'.ssive  divisions.  How  large  are  they  ?  Shall  19  be  tried? 
A\'liat  would  be  true  of  its  quotient?  What  would  be  true  if 
any  number  larger  than  19  should  be  tried?  Is  293  prime? 
How  do  you  know: 

12.  We  may  conclude  that  a  number  is  prime  if  the  suc- 
cessive prime  numbers  have  been  tried  as  divisors  until  the 
quotient  is  less  than  the  divisor,  provided  there  has  been  a 
remainder  after  each  division. 

If  the  divisions  should  be  continued,  the  quotients  would 
be  found  to  be  numl)ers  already  tried  as  divisors.  No  larger 
number  than  the  prime  last  used  can  be  an  exact  divisor,  for, 
if  it  were,  the  quotient  would  be  an  exact  divisor  also ;  but 
this  we  have  seen  to  be  impossible. 


DIVISION.  69 

124.    Exercises  in  testing  Divisibility. 

Tell  whether  or  not  the  following  numbers  are  divisible  by 
three,  and  give  the  reason. 

Illustrative  Example.  45687  is  divisible  by  three  because 
the  sum  of  the  I's  expressed  by  its  digits,  30,  is  divisible  by 
thi'ee. 

1.  6087546.  4.    587642. 

2.  790439.  5.    7300872. 

3.  802563.  6.    14568396. 

Apply  tests  for  4,  5,  6,  on  the  following  numbers,  and 
give  reasons : 

7.  24640.  10.  145380. 

8.  73860.  11.  13060. 

9.  915465.  12.    785630564. 

Apply  tests  for  8  and  9  on  the  following : 

13.  45144.  16.  1046832. 

14.  846396.  17.  5685309. 

15.  4591872.  18.  43549483. 

Apply  test  for  1 1  on  the  following  : 

Illustrative  Example.     604759969. 

The  sum  of  the  I's  expressed  by  the  digits  in  the  odd 
orders  is  33,  and  by  the  digits  in  the  even  orders  is  22. 
The  difference  of  these  sums  is  11,  hence  the  number  is 
divisible  by  11. 

19.  497882605.  22.    24086937. 

20.  6.5834078.  23.    356543847. 

21.  138071406.  24.    958263547. 

Are  the  following  numbers  prime?  Give  the  reason  for 
your  answer. 

25.    887,    941,    767,    1187,    1201,    899. 


70  NEW  ADVANCED  ARITHMETIC. 

125.    Demonstration  of  Tests. 

TEST    FOR    FOUR. 

1.  Any  number  of  more  than  two  orders  may  be  regarded 
as  some  number  of  hundreds  plus  the  number  expressed  by 
the  two  right-hand  figures. 

Since  one  hundred  is  divisible  by  four,  any  number  of 
hundreds  must  be,  by  Principle  1.  If  the  num.ber  expressed 
by  the  two  right-hand  figures  is  divisible  by  four,  the  whole 
number  is,  by  Principle  2. 

Illustrative  Number.     79284  =  79200  +  84. 

79200  is  divisible  by  four,  by  Principle  1.  84  is  divisible 
by  four;  hence,  79284  is  divisible  by  four. 

TEST    FOR    FIVE. 

2.  Any  number  of  more  than  one  order  may  be  regarded 
as  some  number  of  tens  plus  the  number  expressed  by  the 
right-hand  figure.  If  the  right-hand  figure  is  zero,  the  num- 
ber is  tens;  and  since  one  ten  is  divisible  by  five,  the  num- 
ber must  be  divisible  by  five,  by  Principle  2. 

Illustrate  with  a  number. 

TEST    FOR    EIGHT. 

3.  Any  number  of  more  than  three  orders  may  be  regarded 
as  some  number  of  thousands  plus  the  number  expressed  by 
three  right-hand  figures.  Since  one  thousand  is  divisible  by 
eight,  any  number  of  thousands  is  so  divisible,  by  Principle 
1.  If  the  number  expressed  by  the  three  right-hand  digits  is 
divisible  by  eight,  the  whole  number  must  be,  by  Principle  2. 

Illustrate  wuth  a  number. 

TEST    FOR    NINE. 

4.  To  understand  this  test  examine  the  nature  of  tht 
decimal  system. 

10  =      9  +  1.         20  =  (2  X  9)  +  2.         30  ^      27  +  3. 

100  =  99  +  1 .   200  =  2  X  99  +  2.   300  =  297  +  3. 

1000  =  999+1.  2000  r=  2  X  999  +  2.  3000  =  2997  +  3. 


DIVISION.  71 

From  this  partial  tal)le  it  is  evident  that  1  of  any  order 
exceeds  some  multiple  of  9  by  1;  2  of  any  order  exceeds 
some  multiple  of  9  by  2  ;  hence,  we  may  say : 

A  digit  in  any  place  expresses  a  number  that  exceeds  some 
multiple  of  I)  by  as  many  ones  as  the  digit  expresses  when 
standing  alone. 

Since  a  given  number  is  the  sum  of  the  numbers  expressed 
by  its  several  digits  (349  =  300  +  40  +  9),  it  follows  that 
any  number  exceeds  some  multiple  of  9  by  the  sum  of  the 
ones  expressed  by  its  separate  digits. 

If  this  sum  is  a  multiple  of  9,  the  given  num]:)er  is  the  sum 
of  two  multiples  of  9  and  is  therefore  divisible  by  9.  (Prin- 
ciple 2.) 

TEST    FOR    ELEVEN. 

5.    The  following  statements  will  make  this  test  clear : 

(1)  One  ten  is  one  less  than  a  multiple  of  eleven ;  two 
tens  are  two  less;  and  any  number  of  tens  are  as  many 
less  as  there  are  tens. 

A  similar  statement  may  be  made  for  thousands,  hun- 
dred-thousands, ten-millions,  etc.  But  tens,  thousands, 
hundred-thousands,  ten-millions,  etc.,  occupy  orders  whose 
numbers,  counting  from  the  right,  are  even.  Hence,  a  digit 
standing  in  an  order  whose  number  is  even,  expresses  a 
number  which  is  as  many  less  than  a  multiple  of  eleven  as 
the  number  of  ones  expressed  by  the  digit. 

(2)  In  a  similar  manner  it  may  be  shown  that  a  digit 
standing  in  an  order  whose  number  is  odd  expresses  a  num- 
ber which  is  as  many  more  than  a  multiple  of  eleven  as  there 
are  ones  expressed  by  the  digit. 

(3)  If  these  remainders  are  equal,  they  balance  each 
other,  and  the  number  is  a  multiple  of  eleven.  If  one  set 
of  remainders  exceeds  the  other  by  a  multiple  of  eleven,  it 
must  follow  that  the  whole  number  is  a  multiple  of  eleven. 


72  NEW  ADVANCED  ARITHMETIC. 


126.     FACTORING. 

1.  Learn  to  apply  the  test  readily.  Try  the  successive  primes,  begin- 
ning, usually,  witii  2. 

2.  If  the  right-hand  figure  is  0,  the  factors  2  and  5  are  readily  recog- 
nized. In  such  numbers  as  25000,  each  0  implies  2  and  5,  hence  the 
factors  may  be  read  off  at  once : 

2,  5,  2,  5,  2,  5,  5,  5. 

3.  Numbers  ending  in  25,  50,  or  75  may  be  factored  by  inspection,  by 
remembering  that  each  hundred  contains  4  25's. 

Illustration.  1575  =  1500  +  75.  1500  =  60  X  25.  75  ^^  3  X  25.  60  X 
25 +  3X  25  =  63  X  25  =  3X3X7X5X5. 

Write  the  prime  factors  of  numbers  to  100,  iu  the  follow- 
ing form: 

4  =  2X2. 
6  =  2X3. 
8  =  2x2x2.     Etc. 

This  expression  may  be  read,  "4  =  2  times  2,"  or  "the 
prime  factors  of  4  are  2  and  2." 

Learn  the  prime  numbers  to  100  so  that  they  can  be  re- 
peated easily  in  ten  seconds. 

Prime  factor  the  following  : 


(1) 

(-^) 

(3) 

(4) 

102 

201 

301 

400 

105 

203 

304 

403 

108 

217 

310 

407 

120 

221 

319 

427 

125 

247 

323 

437 

150 

2.50 

343 

451 

152 

259 

361 

469 

164 

287 

371 

473 

175 

289 

380 

481 

186 

299 

391 

497 

DIVISION. 

(5) 

(6) 

(7) 

(8) 

500 

650 

825 

10560 

525 

675 

833 

15824 

529 

700 

851 

24860 

539 

703 

869 

55625 

583 

731 

899 

73812 

595 

749 

900 

100000 

600 

767 

917 

121212 

611 

799 

940 

255S50 

629 

800 

950 

640000 

637 

804 

975 

1000000 

73 


127.     CANCELLATION. 

Cancellation  is  a  method  of  shortening  the  work  in  prob- 
lems involving  only  multiplication  and  division. 

PRINCIPLES. 

(1)  Dividing  any  one  of  a  series  of  factors  by  any  num- 
ber divides  their  product  by  that  number. 

(2)  Dividing  dividend  and  divisor  by  the  same  number 
does  not  change  the  quotient. 

1.  Divide  8  X  1)  X  6  by  4 ;  by  9 ;  by  12 ;  by  72. 

Illustration.    ^J^  of  8  X  9  X  6  =  i  of  J  of  8  X  9  X  6.     \  of 
8  X  9  X  6  =  2  X  9'  X  6.     I  of  2  X  9  X  6  =  2  X  3  X  6. 

2.  5x8x0x6  =  ? 

24  X  45  X  60 
Illustrative  Problem.     — i8~x^0 —  ~ 

EXPLANATION. 
This  is  a  problem  in  division  in  which  divisor  and  dividend 
are  partially  factored.  I  shorten  the  operation  by  dividing 
divisor  and  dividend  by  9.  ^  of  the  divisor  is  2  X  20.  ^  of 
the  dividend  is  24  X  5  X  60.  I  further  shorten  the  operation 
by  dividing  divisor  and  dividend  by  4.  ^  of  the  dividend  is 
6  X  5  X  60,  etc. 


74  NEW  ADVANCED  ARITHMETIC. 

12  X  36  X  51 
^'    24  X  18  X  34  -  ' 

4.  Divide  27  x  35  X  52  by  18  X  7  x  13. 

5.  Divide  92  X  87  X  57  X  60  by  23  X  23  X  19  X  29. 

6.  Divide  140  x  169  x  510  by  39  x  68. 

7.  How  many  baskets  of  eggs,  each  contaiuiug  12  dozens, 
at  15  cents,  a  dozen,  mil  pay  for  8  bolts  of  cloth,  each  con- 
taining 24  yards,  at  30  cents  a  yard? 


128.     ANALYSIS. 

1.  If  63  books  cost  8126,  what  will  125  books  cost? 

FOKM. 

S126  x  125 
63 

Analysis.  Since  the  question  asks  for  the  cost  of  certain  articles,  1 
begin  with  $126,  writing  it  above  a  short  horizontal  line.  If  63  books  cost 
$126,  each  book  will  cost  one  sixty-third  of  S126, .which  is  expressed  by 
writing  6.3  below  the  line  as  a  divisor.  125  books  will  cost  125  times  this 
number  of  dollars,  which  is  expressed  by  wTiting  1 25  above  the  line  as  a 
factor  of  the  dividend.  Cancelling  the  common  factors  and  completing 
the  work,  the  result  is  $250. 

2.  If  15  men  can  do  a  piece  of  work  in  7  days,  in  how 
manj'  days  can  21  men  do  the  same  work? 

3.  If  24  men  dig  a  ditch  in  18  days,  how  many  would  be 
required  to  dig  the  same  ditch  in  27  days? 

4.  If  11  tons  of  hay  can  be  made  from  5  acres,  at  the 
same  rate,  how  many  tons  can  be  made  from  65  acres? 

5.  If  12  acres  of  laud  raise  720  bushels  of  corn,  how  many 
acres  would  be  needed  to  raise  1,800  bushels  at  the  same 
rate? 

6.  If  26  horses  cat  a  certain  quantity  of  grain  in  39  days, 
how  many  days  would  it  last  338  horses? 


DIVISION.  75 

7.  If  a  certain  quantity  of  grain  last  46  horses  34  days, 
how  many  horses  would  eat  the  same  amount  in  391  days? 

8.  64  men  can  do  a  piece  of  work  in  57  days,  working  9 
hours  a  day;  in  how  many  days  can  38  do  the  same  work, 
working  8  hours  a  day? 

9.  If  42  men  do  a  piece  of  work  in  18  days,  working  10 
hours  a  day,  how  many  men  can  do  the  same  work  in  90 
days,  working  7  hours  a  day? 

10.  If  91  men  can  do  a  certain  amount  of  work  in  54 
days,  working  9  hours  a  day,  how  many  hours  a  day  must 
162  men  work  to  perform  the  same  labor  in  39  days? 

11.  The  interest  on  what  amount  of  money  at  8  per  cent» 
for  87  days,  equals  the  interest  on  $2,500  at  7  per  cent  for 
261  days? 

12.  At  what  rate  will  the  interest  on  $3,200  for  92  days 
equal  the  interest  on  $4,800  for  46  days,  at  6  per  cent? 

13.  For  how  many  days  must  $965  be  loaned  in  order  that 
the  interest  on  it  at  5  per  cent  shall  equal  the  interest  on 
$2,123  for  125  days,  at  11  per  cent? 

14.  How  many  men  working  12  hours  a  day  will  be 
needed  to  dig  a  ditcli  1,500  ft.  long,  8  ft.  wide,  and  6  ft. 
deep,  in  250  days,  if  36  men  in  180  days  of  9  hours  each  can 
dig  a  ditch  1,080  ft.  long,  9  ft.  wide,  and  12  ft.  deep,  the 
work  to  be  uniformly  difficult? 

For  further  problems  see  Simple  and  Compound  Propor- 
tion. 


76  NEW  ADVANCED  AFdTIIMETIC. 

SECTION    VI. 

FRACTIONS. 
129.    DEFINITIONS. 

1.  A  fractional  number,  or,  more  briefly,  a  fraction,  is  a 
number  that  is  composed  of  one  or  more  fractional  units. 
Illustration.     -|,  ■^^. 

2.  A  fractional  unit  is  one  of  the  equal  parts  into  which 
a  whole  has  been  separated.     Illustration.     ^,  Jj. 

3.  The  numerator  of  a  fraction  is  the  number  of  frac- 
tional units  in  the  fractional  number.  In  the  fractions  |,  |, 
the  numerators  are  2  and  5. 

4.  The  denominator  of  a  fraction  is  the  number  that 
shows  tlie  size  of  the  fractional  unit.  This  it  does  by 
showing  the  number  of  equal  parts  into  which  the  whole 
has  been  separated.  Illustration.  In  the  fractious  f,  y\y 
5  and  12  are  the  denominators.  The  numerator  and  de- 
nominator are  the  terms  of  the  fraction, 

5.  Fractions  are  classified,  with  respect  to  their  value, 
into  proper  and  improper. 

6.  A  proper  fraction  is  a  fraction  whose  value  is  less 
than  one.  Its  numerator  is  less  than  its  denominator. 
Illustration.     |,  j^^. 

7.  An  improper  fraction  is  a  fraction  whose  value  is  equal 
to  or  greater  than  one.  Its  numerator  is  equal  to  or  greater 
than  its  denominator.     Illustration.     |,  f . 

8.  Fractions  are  classified,  with  respect  to  their  form, 
into  simple,  complex,  and  compound. 

-  9.    A  simple  fraction  is  a  fraction  whose  terms  are  inte- 
gers.    Illustration.    ^-,  f. 


FRACTIONS.  77 

10.  A  complex  fraction  is  a  fraction  which  contains  a 

fraction  in  one  or  both  of  its  terms. 

4      4      a      41. 
Illustration.  -,     ^,    ^,    — ^,  etc. 

I      7      4       8 

11.  A  compound  fraction  consists  of  two  or  more  simple 
fractious  joined  b}"  of.     Illustration.     §  of  *. 

12-  A  mixed  number  is  composed  of  an  integer  and  a 
fraction.     Illustration.     A^.    ^ 

13.  Tell  whether  each  of  the  following  is  proper  or  im- 
proper, and  give  the  definition  in  each  case. 

^    ^    ^,  ^    3  of  4    i. 

14.  Tell  whether  each  of  the  following  is  simple,  com- 
plex, or  compound,  and  give  the  definition  in  each  case. 

#5    f>    -if'    i    of   h    i'    o- 
'  4      5 

15.  A  fraction  is  in  its  lowest  terms  when  the  numerator 
and  denominator  are  prune  to  each  other. 

130.    PRINCIPLES. 

I.  Multiplying  the  numerator  of  a  fraction  by  an  integer 
multiplies  the  fraction  by  the  integer. 

Since  the  number  of  fractional  units  in  the  fractional 
number  is  multiplied  by  the  integer,  while  their  size  is  un- 
changed, the  fraction  is  multiplied  by  the  integer.    Illustrate. 

II.  Dividing  the  numerator  of  a  fraction  by  an  integer 
divides  the  fraction  by  the  integer. 

Since  the  number  of  fractional  units  in  the  fractional 
number  is  divided  by  the  integer,  while  their  size  is  un- 
changed, the  fraction  is  divided  by  the  integer.     Illustrate. 

Ill  Multiplying  the  denominator  of  a  fraction  by  an 
integer   divides   the  fraction  by  the  integer. 


78  NEW  ADVANCED  ARITHMETIC. 

If  the  number  of  equal  parts  into  which  a  unit  has  been 
separated  be  doubled,  each  part  will  be  one  half  as  large  as 
before.  If  the  denominator  of  a  fraction  be  multiplied  by 
any  integer,  the  unit  will  l)e  divided  into  as  many  times  the 
number  of  parts  that  it  was  before  as  the  integer  is  times 
one.  The  resulting  fractional  units  will  be  the  same  part 
of  the  former  fractional  units  that  one  is  of  the  integer. 
Since  the  number  of  fractional  units  is  unchanged,  the  frac- 
tion must  be  divided  by  the  integer.     Illustrate. 

IV.  Dividmg  the  denominator  of  a  fraction  by  an  integer 
multiplies  the  fraction  by  the  integer. 

If  the  number  of  equal  parts  into  which  a  unit  has  been 
separated  be  divided  by  an  integer,  the  resulting  fractional 
units  will  be  as  many  times  the  former  fractional  units  as 
the  integer  is  times  one.  Since  the  numerator  is  unchanged, 
the  fraction  is  multiplied  by  the  integer.     Illustrate. 

V.  Multiplying  both  terms  of  a  fraction  by  the  same 
number  does  not  change  its  value. 

Illustration.  ^  =  *.  There  are  4  times  as  many^  frac- 
tional units  in  |  as  in  i,  but  each  is  only  \  as  large. 

VI.  Dividing  both  terms  of  a  fraction  by  the  same  num- 
ber does  not  change  its  value. 

Illustration.  *  =  i.  There  are  only  \  as  many  frac- 
tional units  in  -^  as  in  |,  but  each  is  4  times  as  large. 


131.    REDUCTION. 

Reduction  of  fractions  is  the  process  of  changing  their 
denomination  without  changing  their  value. 

Review  Reduction,  page  6. 

132.  Illustrative  Example.  In  So  there  are  how  many 
quarters  ? 

i^NALYSis.  Since  in  $1  there  are  4  (jiiarters,  in  $5  there  are  5  fours  of 
quarters,  which  are  20  quarters. 


FRACTIONS.  79 

1.  In  $7  there  are  how  many  tenths  of  $1  ?  in  $8?  in  $9? 
in  $16?  in  $86? 

2.  How  many  quarter-yards  in  5  yards ?  in  12  yards?  in 
15  yards?   in  25  yards?   in  63  yards? 

3.  Change  7  to  fifths;  8  to  thirds;  12  to  eighths;  15  to 
sixteenths. 

4.  Tell  how  to  reduce  any  integer  to  an  equivalent  frac- 
tion having  any  denominator. 

133.  Illustrative  Example.  In  $7|  there  are  how  many 
quarters  ? 

Analysis.  Since  in  $1  there  are  4  quarters,  in  $7  there  are  7  fours  of 
quarters,  which  are  28  quarters.  28  quarters  aud  3  quarters  are  31 
quarters. 

PROBLEMS. 
Change  the  following  to  improper  fractions  and  anah^ze 
the  process. 

Note.     Illustrate  Problems  1-3  with  paper  circles. 


1. 

5^. 

6. 

8§. 

11. 

64  f. 

16. 

H- 

21. 

33i. 

2. 

2|. 

7. 

12^. 

12. 

85ig. 

17. 

m. 

22. 

561. 

3. 

3|. 

8. 

15|. 

13. 

91^. 

18. 

41§. 

23. 

66§. 

4. 

H- 

9. 

201*. 

14. 

34/^. 

19. 

26|. 

24. 

81i-. 

5. 

5/.. 

10. 

25|. 

15. 

512^^-r- 

20. 

62i. 

25. 

83|. 

134.    Solve  the  above  problems,  giving  the  following 

Analysis.  5^  =  ?  Since  in  one  there  are  2  halves,  in  any  integer 
there  are  twice  as  many  halves  as  ones ;  hence  in  5  there  are  two  times  5 
halves,  which  are  ^-,  ^^-  +  ^  —_^- 

Show  how  the  following  rule  may  be  made  from  the  above 
analysis. 

RULE. 

]For  reducing  a  tnioced  nuniber  to  an  improper  fraction. 

Multiply  the  integer  by  the  denominator  of  the  fraction. 
To  this  restilt  add  the  numerator,  and  write  the  sum  over 
the  denominator. 


80  NEW  ADVANCED  ARITHMETIC. 


135.    Change: 

1.    8  to  llths. 

7. 

9  to  15ths. 

13. 

18f  to  7ths. 

2.    6f  to  5ths. 

8. 

10  to  18ths. 

14. 

34^7  to  17ths. 

3.    12  to  lOths. 

9. 

llf  to6tbs. 

15. 

46/^  to  25ths. 

4.    12^  to  3ds. 

10. 

15  to  21sts. 

16. 

503?o  to  30tlis. 

5.    7  to  8ths. 

11. 

Uj\  to  lOths. 

17. 

65H  to  12ths. 

6.  7|  to  Sths. 

12. 

17VV  to  leths. 

18. 

72/^  to  18ths. 

136.    Illustrative  Example. 

Change  24  quarters  of  a  dollar  to  dollars. 

Analysis.  Since  in  $1  there  are  4  quarters,  in  24  quarters  of  a  dollar 
there  are  as  many  dollars  as  there  are  fours  in  24.  There  are  6  fours  iu 
24 ;  hence  in  24  quarters  of  a  dollar  there  are  $6. 

Similarly  reduce  the  following  fractions  to  whole  numbers : 


1. 

f,  f,  ¥,  Y- 

6. 

ih 

M,  ¥,  if- 

2. 

hS  ¥,  -VS  ¥• 

7. 

¥, 

¥,  H,  V. 

3. 

V,  ¥,  -¥,  ¥- 

8. 

If, 

-¥,  f^,  H- 

4. 

n,  n,  n,  n- 

9. 

H^ 

¥eS  ^1^ 

W 

5. 

ff,  11,  ih  u- 

10. 

2|5 

W,  ¥o^ 

W 

137.  Illustrative  Example.  Reduce  ^"-  to  a  mixed 
number. 

Analysis.  Since  in  one  there  are  f ,  in  ^  there  are  as  many  ones  as 
tliere  are  fives  in  27.  There  are  54  fives  in  27  ;  hence  there  are  5f  oaes 
in  y. 

Change  the  following  fractions  to  whole  or  mixed  numbers  : 


1. 

¥,  ¥-,  ¥• 

9. 

¥/, 

W,  ¥^- 

2. 

-¥,  ¥,  V- 

10. 

'W, 

¥4S  W- 

3. 

¥,  ¥-,  H- 

11. 

w, 

W,  %¥• 

4. 

n,  ff,  ¥• 

12. 

!M, 

*s^  n§- 

5. 

ih  ff,  !^ 

13. 

Ml, 

fH,  -VW 

6. 

?i,  H.  ?t- 

14. 

%h  ' 

II,  W- 

7. 

W»  W,  -W- 

15. 

W, 

Vif,  W- 

8. 

V^  W>  W- 

16. 

^  3 

,  ^H=^- 

FRACTIONS.  81 

RULE. 
To   reduce  an  improper  fraction   to  a   whole   or  mixed 
number,  divide  the  numerator  by  the  denouilnator. 

17.  7 5  feet  are  how  many  thirds  of  a  foot?  ^^  of  a  foot 
equal  how  many  feet? 

18.  12|  yards  equal  how  many  fifths  of  a  yard?  -<y-  of  a 
yard  equal  how  many  yards? 

19.  $17 1^0  equal  how  many  dunes?  %'^^  equal  how  many 
dollars  ? 

20.  15^  pounds  are  how  many  eighths  of  a  pouud?  ^J*  of 
a  pound  are  how  many  pounds? 

Note.  Give  many  dictation  exercises  until  facility  in  solution  and 
analysis  is  acquired. 

138.  Illustrative  Example.  Reduce  §§  to  its  lowest 
terms. 

Analysis.  Dividing  numerator  and  denominator  by  5,  the  resulting 
terms  are  prime  to  each  other,  hence  the  fraction  is  in  its  lowest  terms. 
I  equals  f §,  by  Principle  6  ;  or,  f  equals  f  §  because  the  resulting  fractional 
units  are  5  times  as  large  as  the  former,  and  there  are  but  ^  as  many  of  them. 

Reduce  the  following  to  their  lowest  terms : 

6.    11,   Ih   %h   If. 
7. 

8. 

9        14i       2«9       _77_5 
^'       180'     30  6,     950* 

RULE. 

To  reduce  a  fraction  to  its  lotvest  terms,  continue  the 
division  of  the  terms  of  the  fraction  by  the  same  num- 
ber until  they  are  prime  to  each  other. 

Express  the  following  in  the  simplest  form : 

11.  fA  of  SI,  $u,  $1^0%'  ^n- 

12.  1^  of  a  yard,  ff  yard,  ^U  yard,  i2§  yard. 

13.  if  of  a  bushel,  ^f  bushel,  ^^  bushel,  |f  bushel. 

14.  4V  of  a  mile,  |^  mile,  ff  mile,  |^  mile. 

6 


1. 

1,  \h  if,  u- 

2. 

§1,  th  If'  if. 

3. 

%h  It,  m  rh 

4. 

ft,    ?|,   M,   T^- 

5. 

th  n,  H'  If- 

82 


^EW  ADVANCED  ARITHMETIC. 


139.    Illustrative  Example.     Change  ^^  to  60ths. 

1.  Analysis.  To  change  -^  to  60ths,  each  12th  must  be  separated 
into  5  equal  parts.  -^  contain  7  times  /j,  which  equals  ff .  Hence,  to 
change  -^^  to  GOths  I  multiply  both  terms  by  5. 

2.  Since  in  one  there  are  f§,  in  ^  there  are  ^^  of  fg.  -^  oi  ^  is  /j. 
^2  of  fa  is  7  X  ^  =  M- 

Is  this  reduction  ascending  or  descending? 


Change : 

1.  f  to  40ths. 

2.  1?^  to  45ths. 

3.  |to56th8. 


PROBLEMS. 

4.  ^  to  36ths. 

5.  ^^  to  80ths. 

6.  A  to  lOOths. 


7.  -jSr  to  55ths. 

8.  ^  to  GOths. 


9. 


12 


to  96ths. 


RULE. 

To  change  a  fraction  to  an  equivalent  fraction  lutrlntj 
any  denominator,  multiply  both  terms  of  the  fraetton  by 
the  quotient  arising  from  diriding  the  desired  denomi- 
nator by  the  given  denominator. 


Note.     Give  many  dictation 

exercises. 

Change 

10. 

f  to  18ths. 

21. 

%\  to  924ths. 

11. 

\\  to  48ths. 

22. 

II  to  343ds. 

12. 

-i"(.  to  96t.hs. 

23. 

/^  to  378ths. 

13. 

Vg  to  120ths. 

24. 

\l  to  860ths. 

14. 

i|  to  lOSths. 

25. 

tVs  to  lOOOths. 

15. 

^1  to  144ths. 

26. 

tVo  to  1350ths. 

16. 

li  to  128ths. 

27. 

\%l  to  2816ths. 

17. 

-i)r  to  108ths. 

28. 

H^  to  2118ths. 

18. 

la  to  333d.s. 

29. 

f  If  to  1563ds. 

19. 

^;^-  to  SGOths. 

30. 

^§fi  to  2.524ths, 

20. 

'il  to  G30ths. 

31. 

£^^  to  26U0th8. 

FRA  CTIONS.  83 

140.     ADDITION    OF    FRACTIONS, 

1.  Define  addition,  sum.  Only  what  numbers  can  be 
united?     Add  i,  |,  |.     Add  J,  f,  4,  f. 

2.  What  is  true  of  the  fractions  in  each  of  these  problems? 
How  is  their  addition  performed  ? 

3.  In  the  problem  §  +  2  =  ?  the  fractional  units  in  the 
first  fraction  are  unlike  those  in  the  second.  Before  these 
two  fractious  can  be  united  they  must  be  made  alike ;  that  is, 
they  must  be  changed  to  equivalent  fractions  having  the 
same  fractional  unit.  The  denominators  wUl  then  be  alike ; 
that  is,  the  new  fractions  will  have  a  common  denominator. 

4.  If  each  of  the  thirds  be  divided  into  two  equal  parts, 
what  are  the  resulting  fractional  units  called?  into  four? 
Name  three  other  fractional  units  that  may  be  made  from 
thirds.  Which  is  the  largest  of  all  these  fractional  units? 
In  the  same  way,  name  successive  fractional  units  that  may 
be  made  from  fourth's.  Have  you  found  any  that  can  be 
made  from  either  thirds  or  fourths?  Wliich  were  they?  f 
equals  how  many  twelfths?  |  equals  how  many  twelfths? 
Tlien  §  +  i  =? 

PROBLEMS. 

2  _i_   r, 

I  +  |. 
1  +  ^- 

It  is  thus  seen  that  to  unite  unlike  fractions  they  must  be 
changed  to  equivalent  fractions  having  a  common  denom- 
inator, hence  we  must  learn  how  to  find  this  common 
denominator. 

141.     LEAST   COMMON   MULTIPLE. 

1.   Let  us  consider  Problem  2  in  Art.  140. 
1  =  ^  =  1  =  ^'^  =  \%,  etc.     These  denominators  contain 
what  common  factor  ?      Every  number  containing  3  as   a 


1. 

h+\- 

5. 

2. 

§  +  f. 

6. 

3. 

i  +  f  . 

7. 

4. 

h  +  h 

8. 

9. 

S  +  i-  +  t- 

10. 

J  +  ^  +  §• 

11. 

h  +  \  +  \' 

12. 

t  +  l  +  i- 

84  NEW  ADVANCED  ARITHMETIC. 

factor  is    a    multiple  of   i).      Name   multiples  of  4,    5,    6, 
8,   10. 

f  =  T%  =  1^5  =  2Mi  *^tc.  These  denominators  are  multiples 
of  5. 

15,  found  in  both  sets  of  denominators,  is  a  multiple  of 
both  3  and  5,  hence  is  called  a  common  multiple.  Name 
two  other  common  multiples  of  3  and  5. 

Name  two  common  multiples  of  2,  3,  and  4  ;  of  3.  6,  and 
9 ;  of  4,  5,  and  8;  of  3,  4,  5,  and  6.  Name  three  common 
multiples  of  2,  4,  and  5 ;  of  3,  8.  and  6 ;  of  4.  5,  and  8. 

2.  A  multiple  of  a  number  is  an  integral  number  of  times 
that  number.  It  is  consequently  exactly  divisible  by  that 
number. 

3.  A  common  multiple  of  two  or  more  numbers  is  a  num- 
ber that  is  a  multiple  of  each  of  them.  It  is  consequently 
divisible  by  each  of  tliem. 

4.  A  mxdtiple  of  a  number  contains  all  of  the  prime 
factors  of  that  number. 

5.  A.  common  multiple  of  two  or  more  numbers  contains 
all  of  the  prime  factors  of  each  of  them. 

6.  How  many  common  multiples  may  two  or  more  num- 
bers have  ? 

7.  Name  three  common  multiples  of  2  and  3  ;  of  4  and  5 ; 
of  3,  4,  and  6.  Which  is  the  least  conamon  multiple  in  each 
case? 

8.  The  least  common  multiple  of  two  or  more  numbers  is 
the  least  number  that  is  a  multiple  of  each  of  them. 

9.  It  is  the  product  of  such  prime  factors,  and  such  only. 
as  are  necessary  to  produce  the  several  numbers. 

142.    Illustrative  Example.     18,  24,  40. 

FORM. 

18  =  2  X  3  X  3  \ 

24  =  2x2x2x3  (2x2x2x5x3x3  =  360. 

40  =  2X2X2X5S 


FRACTIONS. 


85 


Analysis.  The  prime  factors  of  18  are  2  and  two  3's;  of  24  are  three 
2's  and  3  ;  of  40  are  three  2's  and  5.  Since  the  1.  c.  m.  of  these  numbers 
must  contain  40,  it  must  contain  its  prime  factors,  which  I  use  as  factors 
of  the  1.  c.  m.  Since  the  1.  c.  m.  nmst  contain  24,  it  must  contain  also  the 
prime  factor  of  24  not  found  in  40;  lience,  1  use  3  as  a  prime  factor  of  the 
1.  c.  m.  Since  the  1.  c.  m.  must  contain  18,  it  must  contain  also  the  prime 
factor  of  18  not  found  in  40  or  24 ;  hence,  I  use  3  as  a  factor  of  the  1.  c.  m. 
The  product  of  2,  2,  2,  5,  3,  and  3  is  the  I.  c.  m.  of  these  numbers,  because 
it  contains  such  prime  factors,  and  only  such,  as  are  necessary  to  produce 
the  several  numbers. 

EXAMPLES. 


Find  the  1. 

c.  m.  of  the  following  : 

1. 

2,3, 

4. 

16. 

6,  12,  15,  16. 

2. 

3,4, 

6. 

17. 

3,  5,  8,  10,  12 

3. 

4,6, 

9. 

18. 

2,  5,  6,  7,  8. 

4. 

5,6, 

8. 

19. 

6,  7,  8,  9,  10. 

5. 

6,8, 

9,  10. 

20. 

7,  8,  10,  12,  14. 

6. 

4,5, 

6,  8. 

21. 

8,  9,  10,  11,  12. 

7. 

3,5, 

6,  8. 

22. 

8,  12,  20,  30. 

8. 

5,6, 

8,  12. 

23. 

6,  15,  18,  36. 

9. 

4,5, 

6,  7. 

24. 

15,  25,  30,  40. 

10. 

5,  6, 

7,  8,  9. 

25. 

18,  24,  36,  50,  60. 

11. 

6,8, 

9,  10. 

26. 

24,  36,  60,  75. 

12. 

5,  7, 

10,  11, 

12. 

27. 

80,  120,  160,  18. 

13. 

8,9, 

10,  12, 

14. 

28. 

75,  130,  145,  190. 

14. 

7,8, 

10,  12. 

29. 

125,  160,  225,  260 

15. 

8,9, 

12,  15. 

30. 

336,  345,  425. 

143.    Tlie  Inspection  Method. 
niustrative  Example.     Find  1.  c.  m.  of  10,  12,  18,  30,  86; 


45. 


7A 


10 
12 

18 
30 
36 
45  X  2  X  2  =  180,  1.  c.  m. 


8G  NEW  ADVANCED  APdTHMETIC. 

Analysis.  Since  10  is  a  divisor  uf  30,  any  multiple  of  30  contains  10 
I  therefore  strike  out  10.  Similarly  I  strike  out  12  and  18,  divisors  of  35. 
The  largest  factor  of  30  found  in  45  is  15  ;  the  other  factor  is  2 ;  so  the 
1.  c.  m.  must  contain  45  X  2.  The  largest  factor  of  36  in  45  is  9,  the  other 
factor  is  4,  or  2  X  2 ;  heuce  the  1.  c.  m.  is  45  X  2  X  2,  or  180. 


EXAMPLES. 
Find  the  1.  c.  m.  of  the  following : 

1.  8,  16,  30,  48,  60,  75.  9.  27,  38,  45,  54,  60,  76,  90, 

2.  7,  15,  28,  42,  75,  90.  10.  28,  30,  40,  56,  60. 

3.  21,  45,  63,  72,  84.  11.  51,  68,  78,  88,  91. 

4.  28,  44,  56,  70,  88.  12.  23,  46,  69,  92.' 

5.  39,  52,  64,  78,  91.  13.  15,  30,  45,  50,  60,  90. 

6.  42,  58,  84,  91,  98.  14.  3,  7,  13,  25,  49,  56. 

7.  36,  52,  65,  72,  84.  15.  5,  11,  13,  29. 

8.  4,  5,  6,8,  12, 16,  25,  30.  16.  6,  8,  12,  16,  24,  36,  144. 

144.    EXAMPLES. 

-■    ^  +  §  +  J.  12.    f  +  f  +  T^  +  A  +  ^\- 

2.     s  +  I  +  5.  13.    §  +  I  +  i'o  +  H  +  t't- 

o         T       1       1       I       T  1  x  "i       I       4.       r       1  1       I         7 


1 


3.  -2  +  J  +  g.  14.  l  +  ^  +  H  +  1^5- 

4-  *  +  t  +  i-  15.  t  +  tV  +  t'tt  +  t\- 

5.  -^  +  i  +  e  +  tV-  16.  S  +  *  +  i  +  A  +  \h 

6.  i  +  f  +  t  +  ^.  17.  i  +  s  +  ^  +  f  +  i- 

7.  i  +  S  +  ^  +  |.  18.  a  +  f  +  I  +  I  +  tV- 

8.  *  +  g  +   I  +  A-  19-  f  +  i  +  A  +  H  +  ^• 

9.  i  +  t  +  ^  +  t-  20.  t  +  t  +  /ij  +  tV  +  1^5. 

10.  ^  +  i  +  ?  +  t  +  f .  21.  f  +  5  +  5  +  T^. 

11.  I  +  i  +  5  +  A-  22.  S  +  4  +  ^. 


FRACTIONS.  87 

23.  f  +  I  +  T^  +  H-  27.  A  +  j^3  +  ^i  +  i§- 

24.  I  +  f  +  §  +  ii-  28.  ^i  +  iS  +  iil  +  M  +  ^g- 

25.  i  +  §  +  |H-  !  +  I"  29.  -I  +  §  +  ^  +  it  +  A  +  A- 

26.  f  +  t  +  §  +  t\j.  30.  f  +  A  +  i^  +  l^  +  H. 

145.  Short  Method.  ^  +  i  =  ?  Since  the  denominators 
are  prime  to  each  other,  theii-  product  is  their  I.  c.  m.  Since 
the  numerator  is  one  in  each  case,  the  denominator  of  the 
second  becomes  the  numerator  of  the  first  reduced  fraction, 
rfud  the  denominator  of  the  first,  the  numerator  of  the  second. 

RULE. 
To  find   the   stun   of  two  fractions   uJiose  numerators 
are  one,  place  the  sum  of  their  denominators  oier  their 
product. 

EXAMPLES. 

1.  1+1^?  4.    1  +  1-?  7.    J,  +  Jj  =  ? 

2.  1  +  1  =  ?  5.    V.  +  V3  =  ?  8-    ^  +  i  =  ? 

3.  1  +  1  =  ?  6.     T^,  +  J,  =  ?  9.     J  +  ,'n  =  ? 

Is  the  rule  applicable  to  fractions  whose  denominators  are 
not  prime  to  each  other?  In  such  cases  is  the  result  in  its 
lowest  terms? 

Modify  the  preceding  rule  for  such  cases  as  ^  +  |. 

Give  the  results  rapidly  for  the  following  problems : 


1. 

§  +  S  =  ? 

4. 

t  +  f-? 

7. 

f +  ^-? 

2. 

1  +  f  =  ? 

5. 

1^  +  ^  =  ? 

8. 

T%  +  A  =  ? 

3. 

f  +  l  =  ? 

6. 

f  +  A  =  ? 

9. 

tV  +  if^  =  ? 

Note.     Dictate  many  similar  problems 

146.   Practice  adding  small  fractions  mentally  until  facility 
is  acquired. 


NEW  ADVANCED  ARITHMETIC. 


Illustrative  Example,     i  +  |  +  f  +  t^^  +  ;}  =  ? 

FORM. 

i  +  i^  +  I  ^-  g  =  ? 
J  +  i  +  iV  +  ^  +  /;  =  ? 
§  +  ^  +  ^  +  i  +  /j  =  ? 


1. 

2. 
3. 

4. 
5. 


FORM. 

•  9 

32 

9| 

30 

36 

3H 

33 

90 

2 

7-2 
2  5 

147.    In  adding  mixed  numbers  do  not  reduce  tiiem  to  im 
proper  fractions. 

Illustrative  Example.     Add  7§,  9^,  3}^. 


Note.  The  common  denominator 
is  written  at  the  right  of  a  vertical 
line,  tlie  new  numerators  in  column 
for  ease  in  adding.  The  quotient,  2, 
is  written  in  ones'  column. 

91  2  3  ' 

"^36 

1.  2i  +  3i  +  If  +  7|. 

2.  5,;  +  12§  +  4^7^  +  8H. 

3.  12if  +  2611  +  33|  +  865. 

4.  61|  +  124H  +  96i|  +  216J. 

5.  77/4  +  66/5  +  1181  +  45^. 

6.  125^1  +  99f  +  23ia  +  184§. 

7.  286^55.  +  324^  +  78913  +  612^. 

8.  177t\  +  268^^  +  317^  +  4.39|. 

9.  48/g  +  644  +  92^  +  354 y'o- 

10.  291/t  +  37tV  +  G3B. 

11.  Four  rolls  of  carpet  contain  respectively  98|  yd.,  97$ 
yd.,  112|  yd.,  96/^  yd.     What  is  the  total  amount? 


FRACTIONS. 


89 


12.    What  is  the  length  of  border  required  in  papering  this 
room  ? 

Floor  plan.     (Dimensions  given  are  in  feet.) 


lA. 


35> 
•*6 


't 


n 


13.  In  a  box  weighing  12i  pounds,  a  grocer  packed  for 
shipment  ISJ  pounds  of  ham,  |f  of  a  pound  of  tea,  3| 
pounds  of  coffee.  6iV  pounds  of  sugar.  What  was  the  total 
weight  of  the  paclvage  ? 

14.  The  United  States  coins  weigh, —  cent  48  grains,  5-cent 
piece  73^  grains,  dime  SSyV  grains,  quarter-dollar  96^% 
grains,  half-dollar  192^^  grains,  dollar  412i  grains,  quarter- 
eagle  64^  grains,  half-eagle  129  grains,  eagle  258  grains, 
double-eagle  516  grains.  What  is  the  eiitire  weight  of  the 
series  ? 

15.  A  man  has  in  his  purse  2  silver  dollars,  3  half-dollars, 
6  dimes,  and  7  5-cent  pieces.  AVhat  is  the  weight  of  the 
whole?     (Addition.) 


90  NEW  ADVANCED  ARITHMETIC. 

16.  How  many  acres  are  there  in  4  tracts  of  land,  the 
first  containing  885,  tlie  second,  112§  acres;  the  third,  146^ 
acres;  and  the  fourth,  39j'\i  acres? 

17.  P'ind  the  sum  of  12i.\  pounds,  31 GJ  pounds,  518| 
pounds,  2005^  pounds,    and   17^   pounds. 

18.  Find  the  amount  of  coal  in  5  car-loads  weighing  as 
follows:  28|  tons,  '29J  tons,  'dO^Q  tons,  27^§  tons,  and  31| 
tons. 

19.  A  mail  carrier  traveled  12^%  miles  on  Monday,  11§ 
miles  on  Tuesday,  13|  miles  on  Wednesday,  10|  miles  on 
Thursday,  9{'^  miles  on  Friday,  and  14|f  miles  on  Saturday. 
Find  the  whole  distance  traveled  in  the  6  days. 

148.     SUBTRACTION    OF   FRACTIONS. 

1.  Define  subtraction,  minuend,  subtrahend,  remainder. 

2.  Illustrative  Example.     3  —  1  =  ? 

Analysis.  If  |  be  separated  iuto  2  parts,  one  of  which  is  \,  the  other 
will  be  f  or  |. 

7  3   _  -2  11 7_  —  ?  13 8     —   ?  II \  —   1 

i   —  »  —   ■  T2  12  —    •  lo         T5  —    ■  15  18  —    ■ 

When  the  denominators  are  alike,  how  is  the  subtraction 
performed  ? 

3.  I  —  I  =  ?  How  many  eighths  are  needed  to  make  ^? 
to  make  ^?     How  many  eighths  remain? 

4     11 2  .    11 3  .    19  3  •   3  —  _7 

*•       1  a  5120  4  '      20  5)5  12' 

5.  Illustrative  Problem.  Separate  |  of  a  sheet  of  paper 
into  two  parts,  one  of  which  shall  be  |  of  a  sheet. 

This  2  of  a  sheet  of  paper  is  to  be  separated  into  two 
such  parts  that  one  of  them  shall  be  I  of  a  sheet.  Thirds 
are  not  readily  formed  from  fourths ;  hence,  I  change  the 
fourths  into  something  from  which  thirds  may  be  made. 

Fold  the  fourths  together.  Fold  what  you  now  have  into 
three  equal  parts.     Open  the  sheet.     It  is  now  folded  into 


FRACTIONS. 


91 


Fold    the 

heet   into 

four    equal 

parts ;  thus, 


Fold  down 
one  of  the 
fourths ;  thus, 


how  many  equal  parts?  Show  ^  of  the  sheet.  How  many 
twelfths  does  it  contain  ?  Show  |  of  the  sheet.  Now  tear  off 
the  fourth  first  folded  down.  What  part  of  the  sheet  is  left? 
How  many  twelfths  are  there  in  it?  How  many  are  needed 
to  make  §  of  a  sheet?  How  many,  then,  should  you  tear 
off?     What  is  left?     What,  then,  does  |  -  §  equal? 

6.   Application  to  an  abstract  problem. 

i  —  1  =  ?  I  am  to  separate  *  into  two  such  parts  tuat 
one  of  them  shall  be  i.  Since  i  is  not  easily  formed  from 
fifths,  I  change  i  to  twentieths,  from  which  ^  may  be  made. 
J  =  ^B.  ^^  are  needed  to  make  l-  If  ig  be  separated  into 
two  parts,  one  of  which  is  /„,  the  other  is  ^i  ;  hence,  *  —  i 

—  11 

—  2  0' 

Explain  the  following  in  the  same  manner : 


1. 
2. 

3. 
4. 


3    1    — 

5  G    — 

3    2    ^: 


5. 
6. 
7. 
8. 


5  I    — 

7  4    — 

6  _    2    — 


7    


_   1    —  V 


9. 
10. 
11. 
12. 


to  G    —  • 

7      _  4    _  ■? 

T  n  7 

ij  _  a  —  V 

1  J  5    —  • 


14 


_  a  —  ? 


7.  From '  the  foregoing  the  following  analysis  may  be 
derived : 

if  —  f  =  ?  Since  these  fractions  are  unlike,  I  change 
them  to  equivalent  fractions  having  the  I.  c.  d. 


Explain  the  following  problems  as  above 

1.  i  -  S  =  ?  4.     f 

2.  A-^-?  5-    i 

3.  5  -  I  =  ?  6 


I :!   — 
_4     — 

1  1  — 


1^  —  #  =  ? 

13  7  —  • 


92  NEW  ADVANCED  ARITHMETIC. 


7. 

H    -    /2    =    ? 

13. 

n  -  /3  =  ? 

8. 

-B  -  1  =  ? 

14. 

u  -  ^■'o  =  ? 

9. 

^-^  =  ? 

15. 

t§  -  /o  =  ? 

10. 

hi  - 1^3  =  ? 

16. 

M  -  H  =  ? 

11. 

§1  -  H  =  ? 

17. 

M-3I  =  ? 

12. 

ii-U  =  ? 

18. 

gB-ie  =  ? 

Form  a  rule  from  the  analysis  just  given. 
149.    Review  Art.  145.      Make  a  rule  for  problems  like 
the  following.     Give  results  rapidly. 


1. 

i 

-h- 

2. 

h 

1 

4- 

3. 

1 

4 

-h 

4. 

1 
2 

-    I. 

5. 

h 

_    1. 

6. 

i 

-h 

7. 

i 

-tV 

8. 

h 

-h 

9. 

1 

7 

-tV 

10. 

} 

-t\ 

11. 

I 

-h- 

12. 

I 

13. 

1 
1  " 

f    —   T 

14. 

Va-^'o 

15. 

V2-l\l 

16. 

\  -  iV. 

17. 

I'o  -  -I'g 

18. 

5  —  J- 

19. 

2    _    2 

:5         4" 

20. 

n-?. 

21. 

5    —    ? 

22. 

1    —    I- 

23. 

.?  —  rr- 

24. 

1  —  t- 

25. 

1  —  I- 

26. 

2   _   2 
i;          s»  • 

27. 

5    — 

f. 

28. 

i- 

|. 

29. 

l- 

T^(T- 

30. 

3.  _ 
5 

A- 

31. 

3 

7 

A- 

32. 

t- 

1 

7* 

33. 

4    

t\- 

34. 

f- 

A- 

35. 

(T 

f- 

36. 

t- 

t\. 

37. 

f- 

T%. 

38. 

g- 

A- 

39. 

H 

-H 

150.    Illustrative  Problem.     7^  —  4|  =  what? 


()         IG 

n     ^ 

4|         9 

12 

2/2       " 

Analysis.  J  =  ^*2  :  1=^X2-  Since  j*^  is  less  than  ^,  I  take  one  of  the 
7  ones,  leaving  6  ones,  and  reduce  it  to  twelfths.  1  =  ff  ;  H  +  T2  —  rl  > 
H  -  f2  =  A  ;  6-4  =  2.     Hence,  7^  -  4|  =  2^2- 

Note.  That  instead  of  adding  \\  to  /j  I  may  subtract  the  /j  from 
the  H  and  add  the  remainder,  -^,  to  the  i*2>  •'hu^  obtaining  ^,  as  before 


FRACTIONS.  93 

Observe  that  the  process  is  identical  with  that  of  subtrac- 
tion of  whole  numbers. 

Explain  the  following  in  the  same  way : 

1.  17i-8i  =  ?  7.    83^-59|  =  ? 

2.  26a-l9f  =  ?  8.    92H   -  46f  ^  =  ? 

3.  124^0  -  98i  =  ?  9.    I26i|  -  97^  =  ? 

4.  317/g  -  2681  =  ?  10.    532i|  -  48329  =  ? 

5.  91/5  -  48a  -  ?  11-    624if  -  279|5  =  ? 

6.  46Uf-178|^  =  ?         12.    1217^1  -  968i|  =  ? 

151.    ADDITION   AND    SUBTRACTION. 

I.  From  the  sum  of  f  and  /^  take  their  difference. 

3.  The  remainder  is  /^  and  the  subtrahend  |.  What  is 
the  minuend? 

4.  What  must  be  added  to  Gi^  to  produce  lljf? 

5.  A  man  received  $4|  for  butter,  $.5*  for  cheese,  and  an 
amount  equal  to  their  sum  for  vegetables.  He  paid  $3^  for 
sugar,  $4*  for  coffee,  and  an  amount  equal  to  their  differ- 
ence for  tea.     What  amount  was  left? 

6.  85  -  (4Vt  -  Vo)  =? 

7.  4|+  (6ii-2f)=? 

8.  b\l  -  {21  +  IH)  =? 

9.  A  man  bought  160  acres  of  land.  To  A,  he  sold  24^ 
acres;  to  B,  ^1^^  acres;  to  C,  as  much  as  to  A  and  B;  to 
D,  as  much  as  the  difference  between  A's  and  B's.  How 
many  acres  were  left? 

10.  (8}f  + 41) -(2^^.4-34)=? 

II.  3  barrels  contain  156^  gallons  of  oil.  In  the  first  are 
51f  gallons;  in  the  second  49f  gallons.  How  many  gallons 
in  the  third  ? 

12.  Monday  night  the  Mississippi  River  at  St.  Louis  stood 
at  15  feet  above  low-water  mark.     Tuesday  it  rose  1^  feet, 


94  NEW  ADVANCED  ARITHMETIC. 

Wednesday  1^  feet;  Thursday  g  feet;  Friday  it  fell  H  feet-, 
Saturday  2 J  feet.     What  was  its  height  Saturday  night? 

13.  John  is  5§  years  older  than  Thomas.  Thomas  is  2f 
years  younger  than  Harry  and  1  ^  years  older  than  Richard. 
John  is  how  much  older  than  Harry? 

14.  A  pole  22|   feet  long  is  broken  in  two.     One  piece 
is  2|  feet  longer  than  the  other.     What  is  the  length  of  each     ! 
piece  ? 

Query.  How  much  added  to  the  shorter  would  make  it  equal  to 
the  longer  ?     Adding  this  to  the  pole,  would  give  what  total  length  ? 

15.  From  Bloomington  to  Decatur  on  the  Illinois  Central 
railroad  the  distance  is  4o§  miles.  Clinton  is  1^^  miles 
nearer  to  Decatur  than  to  Bloomington.  How  far  is  Clinton 
from  each? 

16.  A  owns  12y'^  acres  of  land;  B  owns  7§  acres  more 
than  A.  -  C  owns  as  much  as  both  A  and  B,  and  D  owns  as 
much  as  the  difference  between  A's  and  B's;  find  B's,  C's, 
and  D's.     Find  the  whole  amount  owned. 

17.  3  of  A's  money  increased  by  *  of  his  money  lacks  ^  of 
his  money  of  being  $955.     How  much  mone}^  had  he? 

18.  The  distance  from  Albany  to  Syracuse  is  148  miles. 
A  starts  from  Albany  for  Syracuse  and  B  from  Syracuse  for 
Albany  at  the  same  time.  A  walks  23|  miles  the  first  day, 
18§  miles  the  second  day,  24j\  miles  the  third,  and  29}J 
miles  the  fourth.  B  travels  in  the  same  time  15 3  miles, 
19.^4  miles,  26^  miles,  and  31^  miles.  Make  diagrams  and 
show  : 

a.  How  far  apart  they  were  at  the  end  of  each  day. 

b.  How  far  each  is  from  the  starting-point  at  the  end  of 
each  day. 

c.  How  far  each  is  from  his  destination  at  the  end  oi  eacn 
day. 


FRACTIOXS.  95 

19.  A  piece  of  cloth  contains  TyV  yards.  What  will  be 
left  after  using  |  of  a  j-ard  for  a  vest,  2^  j-ards  for  a  coat, 
and  2^  yards  for  a  pair  of  pantaloons  ? 

20.  A  farmer  having  7  apple-trees  gathered  from  them  as 
follows:  21  barrels,  3|  barrels,  4f  barrels,  5/^  barrels,  2;* 
barrels,  3§  barrels,  5i  barrels.  He  sold  to  one  man  123 
barrels;  to  a  second,  2^  barrels;  to  a  third  4|  barrels.  How 
many  barrels  were  left? 

21.  In  1834  the  amount  of  gold  in  the  eagle  was  reduced 
from  247^  grains  to  232i  grains.     How  much  was  taken  out? 

22  From  an  ounce  (480  grains)  of  standard  gold  were 
minted  an  eagle,  a  half-eagle,  and  a  quarter-eagle.  How 
many  grains  remained?     (See  Art.  147,  Problem  14.) 

23.  In  1853  the  weight  of  the  dime  was  reduced  from  41^ 
grains  to  38i'2  grains.     How  many  grains  were  taken  out? 

24.  From  a  standard  silver  dollar,  41 2i  grains,  10  dimes 
are  coined?     How  many  grains  of  silver  remain? 

152.    MULTIPLICATION    OF    FRACTIONS. 

1.  Define  multiplication,  multiplicand,  multiplier,  prod- 
uct. 

What  does  the  numerator  show?  What  is  the  effect  pro- 
duced by  multiplying  the  numerator?     Why? 

Unite  five  2's  of  thirds;  six  3's  of  fifths;  seven  5's  of 
eighths.  What  did  you  do  in  each  of  these  cases?  How, 
then,  may  you  multipl}'  a  fraction  b}"  an  integer? 

2  Since  the  numerator  of  a  fraction  expresses  the  numWr 
of  fractional  units  in  the  fractional  number,  it  is  evident  that 
a  fraction  is  multiplied  by  multiplying  its  numerator. 

Note.  The  sis^n  X  was  first  introduced  by  William  Oughtred  iu  1631. 
At  first  the  multiplier  was  uniformly  placed  after  the  sign.  Now  the  mul- 
tiplier frequently  precedes  it. 

The  sign  X  is  read  "  multiplied  by  "  when  the  multiplier  follows ;  as 
7  lbs.  X  5,  9A.  X  |,  f  oz.  X  8  ("  three  fourths  of  an  ounce  multiplied  by 
eight  "). 


96  NEW  ADVANCED  ARITHMETIC. 

The  sign  is  read  "  times  "  when  the  multiplier  preceding  it  is  an  integer 
or  a  mixed  number,  as  5  X  7  lbs.,  6|  X  8  ft.  ("  six  and  three  fourths  times 
eight  feet"). 

The  sign  is  read  "  of  "  when  the  multiplier  before  it  is  a  simple  frac- 
tion; as  I  X  $20  ("three  fourths  of  twenty  dollars"). 

The  sign  is  read  "  b}' "  when  the  factors  are  dimensions;  as,  a  pane 
14"  X  32"  ("a  pane  fourteen  inches  by  thirty-two  inches").  A  door  -3'  — 
8"  XT-  6"  ("  a  door  three  feet  eight  by  seven  feet  six  "). 


153.     THE    MULTIPLIER    AN    INTEGER. 

PROBLEMS. 
1.    Multiply  ^-  by  7c 

Analysis.     Seven  4's  of  fifths  are  %'.     In  \^  there  are  as  many  ones  as 
there  are  5's  in  28.     There  are  5|  fives  in  28  ;  lience  -/  =  5|. 


2. 

5  X  4. 

9. 

T^o  X  y. 

16. 

8jo  X  21. 

3. 

1  X  5. 

10. 

i\  X   11. 

17. 

,15|  X  32. 

4. 

1  X5. 

11. 

1  X   10. 

18. 

26^1  X  13 

5. 

i  X  7. 

12 

A  X  6. 

19. 

9f  X  14. 

6. 

1  X  9. 

13. 

,S  X  9. 

20. 

35H  X  43 

7. 

^x  4. 

14. 

H  X  8. 

21. 

52|  X  49. 

8. 

fi  X  8. 

15. 

7^  X  12. 

22. 

69S  X  56. 

Note.  Observe  that  in  these  problems  the  number  of  fractional  units  ifl 
multiplied  in  each  case. 

154.     PROBLEMS. 

1.    Multiply  t  X  4. 

What  fractional  unit  is  four  times  as  large  as  an  eighth? 
Wliat,  then,  is  4  times  §?  Here  we  multiply  the  size  of  the 
fractional  units,  by  dividing  the  denominator  by  the  integer, 
hence  we  say  "  4  times  ^  is  ^ ;  "  g  is  clearly  4  times  |,  since 
it  has  the  same  number  of  fractional  units  and  they  are  4 
times  as  large. 


2 


FRACTIONS.  97 

5?T  X  7.  14.  ]|  X  14.  26.  Hi  X  72. 

3.  tV  X  8.  15.  i*  X  19.  27.  m  X  64. 

4.  H  X  12.  16.  II  X  34.  28.  ^\\  X  48. 

5.  M  X  13.  17.  H  X  10.  29.  yi  X  38. 

6.  ^i  X  7.  18.  ^i  X  18.  30.  2V5  X  75. 

7.  ff  X  19.  19.  ^5  X  37.  31.  ^^^  X  75. 

8.  H  X  21.  20.  t*3  X  25.  32.  fH  X  64. 

9.  If  X  17.  21.  U  X  19.  33.  3V3  X  49. 

10.  fi  X  26.  22.  \l  X  13.  34.  ||i  X  25. 

11.  H  X  11.  23.  |a  X  27.  35.  //g  X  24. 

12.  43^  X  8.  24.  il  X  29.  36.  78/4  X  28. 

13.  ^^  X  20.  25.  ^1   X  28.  37.  ^^%%^  X  33. 

155.    PROBLEMS. 

Ii  has  been  shown  in  Art.  86  that  the  continued  product 
of  a  multiplicand  and  the  factors  of  a  multiplier  is  the  same 
as  the  product  of  the  multiplicand  and  the  multiplier  itself ; 
thus:  25  X  3  X  2  =1  25  X  6. 

In  such  problems  as  j%  x  10,  we  multiply  the  size  of  the 
fractional  units  by  5,  obtaining  §,  then  the  number  of  frac- 
tional units  by  2,  obtaining  ^3^. 

1.    If  X  12=? 

s  If  =  4  times  3  times  Jf     3  times  |f  =  i^.     4 

12.  II  X  56.  22.  /4L  X  180. 

13.  ^7  X  63.  23.  \l^  X  48. 

14.  |§  X  75.  21.  Ill  X  72. 

15.  ii  X  65.  25.  3V5  X  125. 

16.  |i  X  28.  26.  1-12  X  120. 

17.  li  X  65.  27.  4^5  X  218. 

18.  fi  X  87.  28.  Ill  X  240. 

19.  H  X  100.  29.  f|a  X  250. 

20.  t¥s  X  144.  30.  fig  X  144. 

21.  T%\  X  51.  31.  tVo%  X  245. 


An. 

4.LYSIS.       12 

times 

V^-  = 

'^=8 

2. 

f  X  12. 

3. 

t\ 

X  9. 

4. 

H 

X  28 

5. 

H 

X  24 

6. 

1  .T 

T5 

X  20 

7. 

u 

X  35 

8. 

4? 

X  24 

9 

hi 

X  33 

10. 

l| 

X  40 

11. 

U 

X  45 

98  NEW  ADVANCED  ARITHMETIC. 

"WTiich  of  these  three  methods  may  always  be  employed? 
Which  is  the  most  convenient? 

RULE. 
To  tnultiply  a  fraction  by  an  integer,  divide  the  denomi- 
nator by  the  integer,  ff  possible  ;  if  not  possible,  divide  the 
denominator  by  the  largest  possible  factor  of  the  integer; 
if  the  denoniiutitor  is  not  so  divisible,  nuiltiply  the  nume- 
rator by  the  integer. 

Since  dividing  the  denominator  by  a  factor  of  the  integer 
is  the  same  as  omitting  that  factor  from  the  denominator 
and  the  integer,  the  rule  may  be  shortened  : 

Omit  all  factors  common  to  the  integer  and  the  denomi- 
nator, and  mtiltiply  the  numerator  by  the  remaining 
factor  of  the  integer. 

156.     THE    MULTIPLIER    A    FRACTION. 

PROBLEMS. 

1.  Multiply  8  by  I . 

This  means  find  f  of  8.  Whenever  the  multiplier  is  a 
fraction,  the  problem  may  be  read  in  the  same  way.  There 
are  two  processes  involved:  finding  ^  of  8  (partition),  and 
uniting  three  such  parts  (multiplication). 

Analysis.     ^  of  8  is  2.     |  of  8  are  3  twos,  which  are  6 

Find: 

2.  I  of  12 ;  I  of  15 ;  f  of  21 ;  la  of  28. 

3.  §  of  6  bushels;  §  of  f  ;  |  of  10  cents ;  f  of  |§. 

4.  ^  of  $24  ;   f  of  V- ;  t'u  of  40  acres;  -^^  of  y>. 

In  the  preceding  examples  the  multiplicand  is  first  divided 
by  the  denominator  of  the  multiplier,  and  the  quotient  is 
multiplied  by  the  numerator.  Since  the  order  of  the  opera- 
tions is  immaterial,  we  may  first  multiply  the  muUiplicand 
by  the  numerator  and  divide  this  product  by  the  denomina- 
tor.    Generally  this  will  be  more  convenient;  hence,  the 


FRACTIONS.  99 

RULE, 
To  tnnltiply  by  a  fraction,  tnultiply  by  the  numerator 
and  tliiide  the  product  by  the  denominator. 

5  Multiply  8  by  f ;  9  by  § ;  12  by  f  ;   15  by  f. 

6.  Multiply  21  by  f  ;  24  by  | ;  30  by  | ;  32  by  §. 

7.  Multiply  f  by  f ;  |  by  3  ;  j%  by  § ;  ii  by  g. 

8.  Multiply  ^i  by  f ;  |-|  by  fg  ;  if  by  | ;  H  by  §. 

9.  Multiply  U  by  If;  U  by  H;  tVo  by  H;  HI  by  #f. 

10.  Find  j'o  of  f.     How  find  t^  of  §?     How  y'o  of  §? 

11.  Find  g  of  Vi ;  f  of  j%  ;   §  of  ^  ;   §  of  f . 

12.  Find  f  of  i*i;  f  of  |;   f  of  A;   i  of  |. 

13.  Find  I  ox  t ;   f  of  ^  ;    i%  of  {2  ;   §  of  §§. 

In  these  problems,  how  is  the  numerator  of  the  product 
formed?  the  denominator?  Where  cancellation  is  possible, 
what  should  first  be  done? 

RULE. 
To  }nultij>Iy  a  fraction  by  a  fraction,  cancel  all  factors 
common  to  a  numerator  and  a  denominator,  and  miiltiply 
together  the  reinaininrj  factors  of  the  numerators  for  the 
numerator  of  the  product,  and  the  remaining  factors  of 
the  denominators  for  the  denominator  of  the  product. 


157.     PROBLEMS, 

Multiply : 

1.   H  by  15. 

4.    129  by  fi.         7. 

il  by  f . 

2.   i|  by  48. 

5.    15  by  32.           8. 

1?  by  §f . 

3.    54  by  hh 

6.    ^5  by  |.            9. 

f  of  /,  by  1  of  ^rs- 

10.    §  of  3  of  f 

by  H  of  If  of  A. 

Find  : 

11.       y-  of  560 

12.    §§  of  784. 

13.    f§  of  1872. 

Multiply : 

14.    f  Jf  ^'  by  t 

of  f  of  j%.             15. 

8§  by  6. 

Analysis.     6  times  |  is  4.     6  times  8  is  48.     48  +  4  =  52. 


100 


NEW  ADVANCED  ARITHMETIC. 


Multiply : 

16.  12|  by  10. 

17.  155  by  24. 

FORM. 
Ill 


18.  125^2  by  48. 

19.  584^1  by  50. 


20.  G24^V  by  86, 

21.  Ill  by  81. 


5i 


Use  this  method  wheu  the  fractions  are 
small. 

Analysis.  J  of  11=2  and  a  remainder  of  3. 
3  =  |.  I  +  I  =  ¥•  i  of  i  =  A-  8  X  I  =  5^.  8 
X  11  =  88,  etc. 


96i 

Multiply : 

22.    15|byl2^.      23.  86|by27l.      24.  1248/o  by  492|. 

Reduce  the  mixed  numbers  to  improper  fractions. 

25.  What  is  the  cost  of  ^^  of  an  acre  of  land  at  888  an 
acre? 

26.  What  is  the  cost  of  /-  of  a  quire  of  paper  at  23  cents 
a  quire? 

27.  What  is  the  cost  of  4g  cords  of  wood  at  84.75  a  cord? 

28.  What  is  the  cost  of  23J  yards  of  broadcloth  at  82J  a 
yard  ? 

29.  What  is  the  cost  of  ^^^s  tons  of  coal  at  82.57  a  ton? 

30.  What  is  f  of  I  ?  t  of  /^  of  \\  ? 

31.  What  is  f  of  ^2  of  \\  of  50  ? 

32.  What  is  /^  of  2^  of  \\  of  5 J  of  12? 

33.  Find  the  cost  of  3^  yards  of  cloth  at  83.75;  of  4| 
yards  at  82.40;  of  15f  yards  at  83.20. 

34.  Bought  ^  cords  of  wood  at  83.50  ;  5^  cords  at  84.40  ; 
6|  cords  at  84.56.  Sold  the  wood  at  an  average  price  of 
85.00  a  cord.     What  was  the  gain? 

35.  A  farmer  gathered  42^  loads  of  corn  averaging  33§ 
bushels  from  his  25-acre  fiekL     What  was  the  total  yield? 

36.  Multiply  17|  by  14J;  26g  by  293;  132^^  by  68|; 
694|by87^. 


FRACTIONS. 


lOi 


37.  A  steamer  ran  at  an  average  rate  of  386f  miles  for 
65  successive  days.     What  distance  was  covered? 

38.  Find  tlie  cost  of  469|  bushels  of  wheat  at  603- 
cents. 

£9.  Find  the  cost  of  35|  pounds  of  coffee  at  28f 
cents. 

40.  A  father  worked  6  days  at  S2J  per  day,  his  son  5 
days  at  Sl|,  his  daughter  4  days  at  $| :  what  were  their  total 
earnings  for  the  week  'i 

41.  What  is  the  area  of  a  door  oi  feet  X  7^  feet? 

42.  Draw  a  square  b\  inches  by  5i  inches.  Draw  lines 
dividing  it  into  square  inches.  Multiply  5^  by  5^  and  point 
out  each  partial  product  in  the  diagram. 

43.  The  sheet  on  which  I  now  write  is  7|"  X  lOf"'. 
How  many  square  inches  ? 


Rhomboid. 


44.  A  rhomboid  is  a  four-sided  figure  with  parallel  sides 
and  oblique  angles.  It  was  shown  in  Art.  92  that  the  area 
of  a  rectangle  is  the  number  of  square  units  in  a  row  the 
length  of  the  rectangle  multiplied  by  the  number  of  such 
rows ;   or,  briefly,  area  =  base  X  altitude. 

In  the  rhomboid  the  perpendicular  distance  from  the  base 
to  the  side  opposite  is  the  altitude. 

8A 


102 


XEW  ADVANCED  ARITHMETIC. 


45.    Cut  a   rhomboid  out  of  paper.     Fold  the  base  AB 
upou  itself,  ereasiug  it  along  the  altitude  PK.     Cut  along 

P  K ;  arrange  the 
parts  as  in  the  sec- 
ond figure.  The 
base,  altitude,  and 
area  of  this  rectan- 
gle are  the  same  as 
in  the  rhomboid ; 
hence  area  of  rhom- 


boid 
tude. 


base  X   alti- 


46.    Cut  out  two  equal  paper  triangles,  M  and  N.     Place 
a  pair  of  equal  sides  together  thus,  making  the  rhomboid  B  S. 


Since  the  base  and  altitude  of  triangle  M  and  of  the  rhom- 

.  ^  .       ,        base  X  altitude 
bold  are  equal,  area  of  triangle  =  ■ 

47  Cut  out  five  paper  triangles,  measure  accurately  their 
bases  and  altitudes,  and  determine  their  areas. 

48.  Determine  the  area  of  any  triangular  spaces  in  the 
school-yard.     Draw  each  on  some  convenient  scale. 

49.  Draw  and  calculate  the  area  of  the  following  triangles. 
Use  the  scale  of  one  inch  to  the  foot  in  the  first  set,  one 
quarter-inch  to  the  foot  in  the  second  set. 


FRA  CTIONS. 


l03 


Base. 

50. 

4  ft. 

51. 

2  yd. 

52. 

3f  ft. 

53. 

4tV  ft 

54. 

3ift. 

Altitude. 

2\  ft. 
\\  ft. 
43  ft. 
2§  ft. 

1]  y^i- 


Base.  Altitude. 

55.  20  ft  171  ft. 

56.  18|  ft  61  ft 

57.  93  ft  14|  ft 

58.  6  ft  14^  ft 

59.  4§  yd.  Z-\  yd. 


60.  A  cubic  foot  of  water  weighs  62 J  pounds.  Ice  is  |f 
as  heavy  as  water.  What  is  the  weight  of  a  cubic  foot  of 
ice? 

61.  "What  is  the  weight  of  a  cubic  foot  of  Joliet  limestone 
which  is  2 J  times  as  heavy  as  water?  of  dry  pine  \l  as 
heavy  ? 

62.  A  gallon  of  water  weighs  8 J  pounds.  What  is  the 
weight  of  a  gallon  of  mercury  13|  times  as  heavy?  of  a  gal- 
lon of  milk  which  is  \%%  as  heavy  as  water? 

63.  Standard  silver  is  ^^  pure.  How  many  grains  of 
silver  in  the  standard  dollar  of  412^  grains? 

64.  The  rear  wheel  of  a  bicycle  is  7^  feet  in  circumfer- 
ence, and  revolves  2A  times  as  often  as  the  pedals.  How 
many  miles  are  traveled  in  1,000  revolutions  of  the  pedals? 
(1  mile  =  5,280  feet) 

65.  From  what  number  can  5^|  be  taken  nine  times  with 
no  remainder? 

158.     DIVISION   OF   FRACTIONS. 

1.  Define  Measurement,  Divisor,  Dividend,  Quotient. 

2.  Define  Partition,  Divisor,  Dividend,  Quotient. 

3.  Illustrate  a  problem  in  measurement  and  one  in  parti- 
tion by  using  objects. 

159.    Divisor  an  Integer. 

Illustrative  Problem.     1.    Divide  \^  into  5  equal  parts. 
Is  this  a  problem  in  measurement  or  partition?     Why? 
How  can  it  be  performed  with  objects? 


104  XEW  ADVANCED  ARITHMETIC. 

According  to  the  definitions  heretofore  given,  this  is  a 
problem  in  partition.     It  ma}'  be  read  :    P'ind  I  or  \^  ;    \  of 

2.  Divide  if  by6;  1^  by  3  ;  f  i  by  7  ;  i4byl2;  §lby9; 
U  by  13. 

3.  Divide  II  by  16;  i\  by  17;  |i  by  19;  g|  by  17; 
A\  by  29. 

4.  Divide  fQ  by  15;  fi  by  13;  ff  by  19  ;  ^i^  by  24  ; 
45  by  7;    8?  by  14 ;    5j3^  by  19. 

How  may  all  of  these  divisions  be  performed?  Make  a 
nile  based  upon  the  solution  of  these  problems.. 

Jllastrative  Problem.     5.    Divide  |  by  5. 

How  does  this  problem  differ  from  the  preceding?  In 
what  other  way  ma}'  a  fraction  be  divided  by  an  integer? 
^  of  \^=  -^-Q,  obtained  by  multiplying  the  denominator  by  5, 
which  di^ades  each  fractional  unit  by  5. 

Explain  fully  the  eSect  of  multiplying  'he  denominator 
by  an  integer. 

6.  Divide  ^%  by  4  ;  f  by  7 ;  \\  by  9  ;  2i  by  10  ;  4f  by  13. 

7.  Divide  I  by  8;  §  by  11 ;  f^  by  12;  ^^  by  16. 

8.  Divide  3^  by  7;  5?  by  8;  7f  by  12;  ^  by  9. 

9.  Divide  ^5  by  12. 

An-altsis.     r2  =  3  of  i-     Hence  r2  of  ^  =  i  of  i  of  j^.    i  of  ^^  =  ^. 

1    r>f     2     —     2 
3   01    15  —  ^• 

10.  Divide  j§  by  15;  \1  by  20;  fo  by  25;  |i  by  14; 
§Jbyl8;  Mby  14;  B  by  26. 

11.  Di\ide  |i  by  34  ;  f  §  by  39  ;  ^4  by  36  ;  4^  by  38  ; 
ft  by  85. 

How  was  the  division  performed  in  the  first  set  o.  prob- 
lems? How  in  the  second  set?  How  in  the  third?  ]\Iake 
a  rule  based  on  these  solutions.  "What  cancellations  should 
be  performed? 


FRACTIONS.  105 

160.    Divisor  a  Fraction. 

Illustrative  Problem.   1.    Divide  4  by  5. 

Analysis.  To  divide  4  by  |  is  to  separate  4  into  equal  parts,  each  of 
which  is  f .     4  =  ^/.     lu  \^-  there  are  six  2's  of  tliirds  (or  6  times  |). 

Illustrate  this  problem  by  folding  4  paper  squares  into 
thirds,  and  then  separating  the  thirds  into  groups  of  two 
each. 

2.  Divide  8  by  f . 

8  =  4^".     In  %Q  there  are  20  times  f. 

What  is  done  with  the  integer?     How  is  the  division  per- 
fonned?     Make  a  rule. 
Divide : 

3.  7  by  |.  4.    12  by  g.  5.    10  by  ^. 

RULE. 

To  divide  a  tvhole  ntnnher  by  ti  fraction,  reduce  tJie 
whole  tiuniber  to  the  same  deitoniinfifion  as  the  fraction, 
and  divide  the  numerator  of  the  dividend  by  tlia  mime-' 
fator  of  the  divisor. 

Divide : 

6.  18  by  ^^.  8.    32  by  if.  10.    63  by  U- 

7.  24  by  If  9.    45  by  Jg.  11.   36  by  ||. 

12.  How  many  boxes  each  holding  §  of  a  quart  can  be 
filled  from  8  quarts  of  berries? 

13.  How  many  yards  of  cloth  at  f  of  a  dollar  a  j'ard  can 
be  bought  for  $12? 

14.  If  a  man  can  dig  a  ditch  -^-^  of  a  rod  in  length  in  an 
hour,  how  many  rods  can  he  dig  in  3|  days  of  9  hours 
each? 

15.  At  2^  cents  each,  how  many  apples  can  be  bought  for 
52  cents? 

Reduce  the  mixed  number  to  an  improper  fraction      2^  -~  ^. 


106  NEW  ADVANCED  ARITHMETIC. 

16.  At  $21  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  863  ? 

17.  At  83^  a  day,  how  many  days  must  a  man  work  to 
earn  $144? 

18.  At  S24|  an  acre,  how  many  acres  can  be  bought  for 
$724? 

19.  If  one  horse  cost  $124[^,  how  many  horses  can  be 
bought  for  83,990? 

161.     Illustratice  Problem.     1.    Divide  1  by  §. 

Analysis.  I  =  |-  Iu  f  there  are  as  many  f  as  there  are  2's  in  5.  The 
quotient  of  5  by  2  may  be  expressed  thus :  |.     Hence  1  -^  |  =  |. 

Explain  the  following  in  the  same  way : 
Divide  1  by  §  ;  by  f ;  by  f  ;  by  ^ ;  by  U  ;  by  ii  ;  by  if ; 
byH;  byf-^  byn- 

Note.  The  denominator  is  the  number  of  fractional  units  in  1.  The 
numerator  is  the  number  of  fractional  units  in  the  fraction ;  hence  the 
rule. 

RULE. 
To  divide  1  by  a  /faction,  divide  the  deuoininator  of  the 
fraction  by  its  niinierator. 

Note.     This  is  usually  called  "  inverting  the  divisor." 

2.    Divide  15  by  f. 

Analysis.  If  the  dividend  were  1,  the  quotient  would  be  |.  Since 
the  dividend  is  15,  the  quotient  is  15  times  I,  which  ecjuals,  etc. 

Analyze  Problems  2  to  11  (Art.  160)  by  this  method. 

162.    Division  of  a  Fraction  by  a  Fraction. 

Illustrative  Problem.    Divide  ^  by  ^. 

Fold  a  paper  square  into  fourths.  Tear  out  one  of  the 
fourths.  We  are  now  to  see  how  many  pieces,  each  of  which 
is  one  third  of  the  paper  square,  can  be  made  from  the  three 
fourths.  Since  thirds  are  not  easily  made  from  fourths,  we 
change  the  three  fourths  to   nine  twelfths.     Four  twelfths 


FRA  CTIOXS.  107 

make  one  third.  Nine  twelfths  make  two  thirds  with  one 
twelfth  left,  which  is  one  fourth  of  another  third.  Hence, 
in  i^  of  a  sheet  of  paper  there  are  2\  thirds  of  a  sheet. 

Explain  the  following  problems  by  the  same  method. 

Divide : 

1.  |byi.  3.    gby|.  5.    §  by  ^. 

2.  §byi.  4.    Aby|.  6.    fbyi. 
Now  omit  the  reference  to  objects.     What  do  you  do  with 

divisor  and  dividend? 

RULE. 
To  divide   a    fraction    by   a    fraction,    change    them   to 
equivalent  fractions    having    the  I.   c.  d.,  and   divide    the 
numerator    of    the    dividend    by    the    numerator    of    the 
divisor. 

7.  Divide  A  by  i. 

Analysis.     ^  =  H-    ^  -  f§.     ?§  4-  f§  =  63  -  10  =  f§  =  6^^. 

8.  Divide  i  by  t- 

Analysis.    ^  =  3^.     f  =  ff .     5%  ^  |§  =  6  -  25  =  ^%. 

9.  Di^^de  ^\  by  f  ;  ^%  by  f ;  %  by  ^  ;   Yi  by  |. 

10.  Divide  21  by  If;  3}  by  2^ ;   1^  by  3|. 


163. 

Shorter  Analysi 

1. 

Divide 

5  by  |. 

Analysis. 

1-1  =  1- 

i  -f  = 

1  of  1  =  i|. 

2. 

Divide 

S  byf; 

1  byl 

f  by  1; 

by  a ;  f  by  v  5 


I  by|i;  HbySI. 
Cancel  common  factors. 


RULE. 
To  divide  a  fraction    by  a  fraction,  invert  the  divisor 
and  proceed  as  in  multiplication. 

Use  the  following  form,  ^  -H  /tt  =  M  X  ¥  =  if  • 
3.   Divide  U  by  /o  ;  I?  by  ^\ ;  H  by  il- 


108  NEW  ADVANCED  ARITHMETIC. 

164.      PROBLEMS. 

1.  At  $f  a  pound,  how  many  pounds  of  coffee  can  be 
bought  for  Soi? 

2.  If  a  man  travel  43  miles  an  hour,  in  how  many  hours 
will  he  travel  23 1  mUes? 

3.  At  82-^  a  yard,  how  manv  vards  of  cloth  can  be  bousht 
forSGi? 

4.  If  each  bag  hold  If  bushels,  how  many  bags  will  be 
needed  to  hold  40 ^  bushels  of  oats? 

5.  At  Sfy  a  yard,  how  much  cloth  will  8f  buy? 

6.  If  an  acre  of  land  will  yield  23/^  bushels  of  wheat, 
how  many  acres  are  necessary  to  yield  1,874*  bushels? 

'•  a  - 1)  -  (f  X  I)  =  ? 

8-  (!  +  *)-^(t-|)  =  ? 

9-  §  of  f  of  f\  -^  i  of  i  of  185  =  ? 

10.  (^  X  if)  -  m  -  i;  =  ? 

11.  Divide  il  by  9  ;   if  by  25  ;   /^  by  10. 

12.  Divide  6  by  * ;   12  by  f ;   15  by  |. 

165.    COMPLEX    FRACTIONS. 

Problems  in  division  of  fractions  are  sometimes  written 

in  the  form  of   a  fraction ;   thus,  5  -f-  J  may  be  written  -|- 

4 
Such  expressions  are  called  Complex  Fractions. 

1.  The  following  method  of  reading  complex  fractions  is 
recommended.  The  complex  fraction  whose  numerator  is  | 
and  denominator  §. 

Read :  2^  § 

5.  2_  .|_  f  5"  J^  4 

V  r  7'  V  8^'         4i'  6- 

2.  The  longest  straight  line  used  separates  the  numerator 
from  the  denominator.     This  line  mav  be  regarded  as  a  sign 


FRACTIONS.  109 

of  division..     The  expression  above  it  is  the  dividend,  and 
that  below  it  the  divisor.     Solve  the  following  as  problems 


in  division. 

Reduce 

1. 

3 

4 

5. 

.5 

2. 

91 

-3 

6. 

8-§ 

3. 

% 

12' 

7. 

8  X  J 
^  X  8 

4. 

1  of 

fof 

9 
TO 

8. 

10. 

11. 

12. 


-A 

f  of  2i 

91    - 
-^3 

^f 

§x 

6i 

H- 

-1 

IX 

9§ 

92  —  J 


166.    To  Find  the  Part  which  One  Number  is  of  Another. 

1.  3  is  what  part  of  7  ? 
Analysis.     1  is  ^  of  7,  hence  3  is  f  of  7. 

2.  5  is  what  part  of  12?  6,  of  17?  11,  of  22?  5,  of  20? 
8,  of  24?  9,  of  36?  17,  of  12?  19,  of  7? 

The  part  which  one  number  is  of  another  is  alwa^'s  ex- 
pressed by  a  fraction,  of  which  the  number  that  is  the  part 
is  the  numerator,  and  the  other  the  denominator. 

Make  a  rule  for  finding  the  part  that  one  integer  is  of 
another. 

3.  §  is  what  part  of  .5? 

Analysis.  5  =  ^f.  |  is  the  same  part  of  ^  that  2  is  of  15.  2  is  ^ 
of  15  ;  hence  f  is  ^  of  5. 

4.  I  is  what  part  of  8?  |  is  what  part  of  12?  |  is  what 
part  of  10?  1  is  what  part  of  9?  j^  is  what  part  of  8?  j\ 
is  what  part  of  6  ? 

5.  1^  is  what  part  of  4?  -J-§  is  what  part  of  14?  \l  is  what 
part  of  126? 

6.  2^  is  what  part  of  6  ?  3  j  is  what  part  of  4  ?  5|  is  what 
part  of  10? 


110 


NEW  ADVANCED  ARITHMETIC. 


Change  the  mixed  numbers  to  improper  fractions. 
Make  a  rule  for  finding  the  part  that  a  fraction  is  of  ao 
integer. 

7.  I  is  what  part  of  |  ? 

Analysis.  |  =  f^.  f  -  |§.  f^  is  the  same  part  of  f§  that  24  is  of 
25.     24  is  II  of  25 ;  hence  -f  is  f |  of  f . 

What  was  done  in  the  above  problem  ?  Mrke  a  rule  for 
such  cases. 

8.  f  is?  what  part  of  f  ? 

9.  I  is  what  part  of  §  ? 

10.  i\  is  what  part  of  i  ? 

11.  With  each  of  the  following  pairs  of  numbers,  find  the 
part  which  the  first  is  of  the  second,  and  give  the  results 
rapidly. 


(1)  h  h 

(9)  h  h 

(17)   7t3„  8f. 

(2)   h  h 

(10)   h  h 

(18)   8^,  lOf 

(3)  1,  i. 

(11)  4|,  51. 

(19)   12^3^,  15  J. 

(4)  h  h 

(12)   J,  81. 

(20)   19i,  14|. 

(5)  h  6. 

(13)   i,  9. 

(21)  211,  25|. 

(6)   1,  8. 

(14)   9,  i. 

(22)   H,  T^o- 

(7)  8,  61. 

(15)  f,|. 

(23)   H,  H- 

(8)  6^,  8. 

(16)   5§,  4|. 

(24)  22^1^,  455»j 

Show  how  the  following  general  rule  is  derived : 

To  find  the  part  that  one  nnniher  is  of  another,  divide 
the  number  ex-pressing  the  part  by  the  number  of  which 
it  is  a  part. 


167.    To   Find  a  Number  when  a   Specified  Part  of  it  is 

Given. 

Illustrative  Problem.  15  is  ^  of  what  number?  Since  15 
is  I  of  the  required  number,  ^  of  that  number  is  J^  of  15.  ^ 
of  15  is  5.  I  of  the  required  number  is  8  fives,  which  are  40. 
Heuce  15  is  §  of  40. 


FRACTIONS. 

1 

Find  the  number  of  w 

lich 

1.    45  is  f . 

8. 

125  is  U. 

ls. 

18|  is  ^. 

2.    48  is  3. 

9. 

324  is  ^|. 

16. 

413  is  ^^. 

3.    36  is  j%. 

10. 

441  is  |i. 

17. 

462  is  ^5. 

4.    72  is  if. 

11. 

|is|. 

18. 

TT  is  1%' 

5.    75  is  If. 

12. 

2  ia   •*> 

3  1°    jr. 

19. 

91§  is  H- 

6.    84  is  ^^. 

13. 

3|  is  /5. 

20. 

23|  is  -If. 

7.    90  is  \l. 

14. 

7|  is  il- 

21. 

34A  is  U 

111 


22.  A  has  §3^  and  B  $7*.  A's  money  is  the  same  part 
of  B's  that  B's  is  of  C's.     How  mucn  has  C? 

23.  A  farmer  has  15f  acres  of  meadow  and  40  acres  of 
oats.  The  part  which  the  meadow  is  of  the  oats-field  is  the 
same  that  the  latter  is  of  the  corn-field.  How  many  acres  of 
corn  has  he? 

24.  A  house  cost  §  as  much  as  the  lot.  Both  cost  $896. 
Find  the  cost  of  each. 

25.  The  area  of  North  America  is  6,446,000  square  miles. 
This  is  about  §  of  the  area  of  Africa.  Whi.t  is  the  approxi- 
mate area  of  the  latter? 

26.  The  area  of  Australasia  is  3,288,000  square  miles. 
It  is  about  what  part  as  large  as  North  America?  as  Africa 
(11,514,000)?  as  South  America  (6,837,000)? 

27.  The  annual  expenditure  of  England  per  capita  for 
military  purposes  is  S3. 72,  and  for  education  is  70  cents  per 
capita.  The  latter  is  what  part  of  the  former?  (Approxi- 
mate.) 

28.  The  distance  from  one  corner  of  a  field  to  another 
corner  is  770  yards;  this  is  y^^  of  a  mile.  How  many  feet 
are  there  in  a  mile? 

29.  A  certain  farm  contains  340  acres  which  are  ^|  of  a 
section.     How  many  acres  are  there  in  a  section? 

30.  A  merchant  sold  goods  to  the  amount  of  $316.80  on 
Monday;  this  was  ^^  of  his  sales  on  Tuesday,  which  were 


112  NEW  ADVANCED  ARITHMETIC. 

*  of  his  sales  on  Wednesday.     What  was  the  aggregate  of 
his  sales  for  the  three  days? 

31.  In  a  school-room  there  are  three  windows  each  9  feet 
X  3i.  If  the  lighting  area  is  2^^  of  the  floor  area,  what  is  the 
latter?     If  the  room  is  30  feet  wide,  what  is  its  length? 

168.     MISCELLANEOUS    PROBLEMS. 

Oral  Exercises. 

(Use  no  written  work  in  the  solution  of  these  problems.) 

1.  Find  the  1.  c.  m.  of  4,  5,  6 ;  of  8,  12,  20;  of  5,  15,  30, 
40;  of  4,  6,  8,  10,  12,  15;  of  12,  15,  18,  20,  30. 

2.  Find  the  1.  c.  m.  of  12,  18,  24;  of  15,  30,  40,  45;  of 
18,  24,  30,  36;  of  60,  90,  105,  120;  of  80,  120,  160,  240. 

3.  Change  to  whole  or  mixed  numbers  ->/,   ^,  ^^a,  f|, 

L25     JL60 
19    »      24    • 

4.  Change  to  improper  fractions  9f,  lO^Sj-,  I2f ,  15f,  18§, 
21i. 

5.  Add  i  and  1 ;  f  and  f ;  |  and  | ;  |  and  ^ ;  §,  |,  and  |; 
I,  ^,  i^o,  and  tV- 

6.  |-i=?     f-|=?     |-f=?     A-  +  =? 

7.  Multiply  f  by  4 ;  /^  by  24 ;  ^^  by  45 ;  ^6^  by  44 ;  ^^ 
by  8. 

8.  Find  f  of  16 ;  ^^  of  36  ;  ^\  of  10:  ^^  of  13  ;  jf  of  7. 

9.  Find  §  of  I;  t  of  if  ;  /^  of  T-V;  A  of  ^^. 

10.  Multiply  t  by  f  ;  2^  by  3^  ;  6^  by  2i ;  f  by  ^^ ;  H 

by?,. 

11.  Divide  4  by  ^;  7  by  i;  lObyi;  12byTir;  6  by  §; 
7  by  1% ;  8  by  | ;  6  by  H;  15  by  ^^ ;  14  by  §  ;  18  by  f  ;  20 
by  5;  lOby  f;  BbyU;  8  by  H;  7  by  H- 

12.  Divide  i¥  by  5  ;  H  by  6  ;  /^  by  16  ;  i\  by  18  ;  j§  by 
36;  H  by  48;  H  by  51 ;  ^  by  87;  ff  by  39;  f|  by  72. 


FRACTIONS.  113 

13.  Divide  1  by  i;  1  by  i;  1  by  ^V ;  1  by  §;  1  by  t\; 
Ibyfg;  1  byH;  1  by  A^o- 

If  one  be  divided  by  any  fraction,  what  will  tlie  quotient 
be? 

14.  Divide  §  by  t ;  t  by  f ;  |  by  ^V  ;  H  by  f  §  ;  ^  by  6f . 

15.  t-ft=?  t-f-?  tXf=.?  1^1-?  f  is 
what  part  of  |  '^     |  is  what  part  of  ^  ? 

16.  7  is  what  part  of  13?  8,  of  19?  11,  of  44?  18,  of  27? 
§  is  what  part  of  6  ?  of  9  ?  of  15  ?  ^  is  what  part  of  4  ?  of  7  ? 
of  10?  of  12?  of  15? 

17.  I  is  what  part  of  *  ?  of  -i%  ?  of  f  ?  of  |§  ?  /„  is  what 
part  of  I'i  of  t^^?  of  \V  of  2^?  of  3J? 

18.  14f  is  f  of  what  number? 

19.  I  of  15  is  f  of  wliat  number? 

20.  I  of  I  of  15  is  I  of  i  of  9  times  what  number? 

21.  f  of  21  is  I  of  what  number? 

22.  I?  of  m  is  ^Q  of  5  times  what  number? 

23.  John  has  ^§  of  a  dollar;  William  has  f  as  much,  and 
this  is  16  cents  more  than  |  of  Henry's  money.  How  much 
has  Henry? 

i24.  John  lost  f  of  his  marbles,  and  has  15  left.  How 
many  had  he  at  first? 

25.  Mary's  money  is  f  of  Laura's,  and  both  have  90 
cents.     How  much  has  each? 

26.  A's  farm  is  |  of  f  of  B's.  A  and  B  together  own  81 
acres.     How  many  acres  has  each? 

27.  John  is  7;  Fred  9  :  (1)  The  difference  in  their  ages  is 
what  part  of  John's  age?  (2)  Of  Fred's  age?  (3)  John's 
age  is  what  part  of  Fred's  age?  (4)  John's  age  is  what  part 
of  Fred's  age  less  than  Fred's  age?  (5)  Fred's  age  is  what 
part  of  John's  age  more  than  John's  age? 

Note.  In  concrete  problem.s,  or  where  a  fraction  expresses  the  relation 
between  two  numbers,  "  of  "  follows  the  fraction.     If  one  number  is  a  frao- 


114  NEW  ADVANCED  ARITHMETIC. 

tion  of  a  second  number  more  or  less  than  the  second  number,  the  "  of" 
phrase  is  frequently  omitted  ;  thus  Jane's  age  is  \  less  than  Mary's  means, 
"Jane's  age  is  \  of  Mary's  less  than  Mary's." 

28.  A  raised  150  bushels  of  potatoes,  which  was   \  less 
than  what  B  raised.     How  many  did  B  raise? 

29.  James  has  884,  which  is  \  less  than  B's  money.     How 
much  has  B  ? 

30.  A  walked  120  miles,  which  is  i  more  than  B  walked, 
and  i  less  than  C  walked.     How  far  did  B  and  C  walk? 

31.  63  is  1   more  than  what   number?  It  is  \  less  than 
what  number? 

32.  The  time  past  noon  is  ^   of  the  time   till   midnight. 
What  o'clock  is  it  ? 

33.  The  time  till  midnight  is   V  of  the  time  past  noon. 
Wliat  o'clock  is  it? 

34.  The  time  till  midnight  is  f  of   the  time  past  noon. 
What  o'clock  is  it? 

At  what  hour  is  the  time  till  midnight  |  of  the  time  past  3 

A.  M.? 

35.  How  many  cubic  inches  in  a  brick  8"  X  4"  x  2\  "? 
How  many  half-inch  cubes  in  a  two-inch  cube? 

36.  Arrange  the    fractions    f,    \\^    j\,    ^|,  in   order   of 
magnitude. 

37.  I  of  water  is  oxygen.     What  weight  of  oxygen  in  a 
gallon  of  water  weighing  8 J  pounds? 

38.  ^  of  air  is  oxygen.     How  much  oxygen  in  a  cubic 
foot  of  air  weigMng  52.5  grains? 

39.  If  15  gold  pens  cost  S25.  what  is  the  cost  of  3  gold 
pens? 

40     If  21  sheep  are  worth  856,  what  are  3  sheep  worth? 

Query.     Is  it  necessary  to  find  the  cost  of  one  shfep? 

41.   If  24  men  consume  a  barrel  of  flour  (196  pounds)  in  2 
weeks,  how  many  pounds  do  3  men  consume  ? 


FRACTIONS.  115 

42.  If  32  horses  in  3  days  consume  60  bushels  of  oats, 
how  many  bushels  do  20  horses  consume  in  one  day  ? 

43.  If  12  bushels  of  oats  are  worth  9f  bushels  of  corn, 
10  bushels  of  oats  are  worth  how  much  corn? 

44    If  8  barrels  of  flour  cost  $25^,  what  do  11  barrels  cost? 

45.  If  at  noon  on  a  certain  day  a  15-foot  pole  casts  a  12- 
foot  shadow,  what  length  of  shadow  is  cast  by  a  pole  32  feet 
long  ? 

46.  A  pine  block  3  "  X  3  "  X  3  "  weighs  5  ounces.  What 
is  the  weight  of  a  similar  block  3 "  X  4  "  X  6  "  ? 

47.  A  can  dig  a  ditch  in  6  days.  What  part  of  it  can  he 
dig  in  one  day? 

B  can  do  the  same  work  in  4  days ;  what  part  can  both 
dig  in  one  day  working  together? 

48  A  and  B  can  trim  2%  of  a  hedge  in  one  day.  In  what 
time  can  they  trim  .^l  of  the  hedge?  fg  of  the  hedge? 

49.  A  can  do  a  piece  of  work  in  7  days.  A  and  B  can  do 
the  same  work  in  4  days.  In  what  time  can  B  do  the  work 
alone  ? 

50.  A  can  mow  a  field  in  5  days,  and  B  can  mow  it  in  6 
days.  In  how  many  days  can  they  mow  it,  working 
together  ? 

51.  A  can  perform  a  certain  work  in  8  days,  B  in  10  days, 
and  C  in  12  days.  In  what  time  can  the  three  perform  it, 
work  i  ng  together  ? 

52.  Three  pipes.  A,  B.  and  C,  fill  a  cistern  in  3  hours.  A 
alone  fills  it  in  8  hours;  B  in  12  hours.  In  what  time  can 
C  alone  fill  it? 

53.  Pipe  A  can  fill  a  cistern  in  10  hours;  pipe  B  in  12 
hours;  pipe  C  empties  it  in  1.5  hours.  When  all  three  pipes 
are  open,  how  long  is  the  cistern  in  filling? 

54    A  can  saw  a  cord  of  wood  in  ^  a  day,  B  in  ^  of  a  day. 
In  what  time  can  both  saw  it  working  together? 
Query.     How  much  does  each  saw  in  a  day  i 


116  NEW  ADVANCED  ARITHMETIC. 

55.  John  can  spade  a  garden  in  J  of  a  day,  Charles  in  *  of 
a  day.     In  what  time  can  they  do  it,  working  together? 

56.  If  I  of  a  yard  of  cloth  cost  81*,  what  will  j\  of  a 
yard  cost? 

57.  If  to  I  of  the  cost  of  an  article  $2  be  added,  the 
result  will  be  f  of  the  cost;  what  is  the  cost? 

58.  J  of  the  distance  from  Chicago  to  Elgin  is  -f^  of  the 
distance  from  Chicago  to  Aurora.  Elgin  is  4  miles  farther 
from  Chicago.     How  far  is  each  from  Chicago? 

Analysis.  f  of  dist.  to  E  =  jSg  of  dist.  to  A. 

^  of  dist.  to  E  =  1%  of  dist.  to  A. 
Dist.  to  E  =  f^  of  dist.  to  A. 
Dist.  to  A  =  i|  of  dist.  to  A. 
Difference  of  di.^tances  =  ^  of  dist.  to  A. 
Difference  of  distances  =  4  miles. 
i\  of  dist.  to  A  =  4  miles. 
^  of  dist.  to  A  =  2  miles, 
f^  of  dist.  to  A  =  42  miles,  dist  to  E. 
jl  of  dist.  to  A  =  3S  miles,  dist,  to  A. 

59.  If  f  of  the  distance  from  C  to  D  is  |  of  the  distance 
from  E  to  F,  and  if  the  sum  of  the  two  distances  is  82  miles, 
what  is  the  distance  from  C  to  D?  from  E  to  F? 

60.  A  house  and  lot  cost  85,200.  The  lot  cost  yV  as  much 
as  the  house.     What  did  each  cost? 

61.  IJ]  is  f  of  what  number? 

62.  What  is  the  cost  of  6^  yards  of  cloth  at  82i  a  yard? 
of  4 1  yards  at  $2^  a  yard?  of  75  yards  at  81-4^  a  yard? 

63.  How  many  pounds  of  material  can  be  bought  for  85^ 
at  8^  a  pound?  for  84f  at  8v\  a  pound?  for  88f  at  8/^  a 
pound?  for  810^  at  82  a  pound? 

64.  If  A  had  84  and  lost  8|,  what  part  of  his  money  did 
he  lose?  if  he  had  85  and  lost  ^?  if  he  had  86^  and  lost 
84?  if  he  had  812^  and  lost  87?  if  he  had  810i  and  lost 
S6i? 


FRACTIONS.  117 

65.  What  is  the  cost  of  i-  a  yard  and  \  of  a  yard  at  $f  a 
yard?  of  §  +  §  at  $§  ?  of  |  +  f  at  %,\  ?  of  \  +  |  at  $§  ? 

66.  A  grocer  bought  equal  lots  of  eggs  at  9  cents  per 
dozen  and  10  cents  per  dozen.  He  sold  them  at  12  cents  per 
dozen,  clearing  60  cents.     How  many  dozen  did  he  buy? 

67.  Sarah's  age  is  §  of  Mary's  and  %  of  Ruth's.  The 
sum  of  their  ages  is  46  years.     How  old  is  each? 

IsOTE.     1  iud  au  expressiou  for  each  iu  sixths  of  iSarah's  age. 


WRITTEN     PROBLEMS. 

1.  The  product  of  three  numbers  is  124:2.  Xwo  of  the 
numbers  are  7 J  and  8].     What  is  the  third? 

2.  What  number  divided  by  %  of  ^2  equals  132§? 

3.  What  nimiber  diminished  by  2.53y'3  leaves  84y  ? 

4.  What  is  the  1.  c.  m.  of  60,  125,  180,  225,  250? 

5.  Bought  8^  yards  of  cloth  at  S3.\  a  yard,  12-;  yards  at 
$2 1,  and  18^  yards  at  $4^.  How  many  bushels  of  corn  at 
26|  cents  a  bushel  will  pay  the  bill? 

6.  15  X  A  X  ^*  X  1068=? 

7.  What  is  the  cost  of  28  pounds  of  butter  at  37  ^  cents  a 
pound,  84  bushels  of  corn  at  43^^  cents  a  bushel,  135  bushels 
of  oats  at  25  cents  a  bushel,  160  bushels  of  rye  at  62^  cents 
a  bushel  ? 

8.  A  man  performed  y^s  of  ^I's  journey  the  first  day,  jAj  of 
it  the  second  day.  §  of  it  the  third  day,  and  had  17g  miles 
left.     What  was  the  length  of  his  journey? 

9.  A  man  left  f  of  his  estate  to  his  eldest  son,  ^-^  of  it  to 
his  second  son,  and  the  remainder  to  his  third  son.  The 
share  of  the  second  was  S520  less  than  that  of  the  third. 
What  was  the  value  of  the  estate  ?  What  was  each  son's 
share  ? 

9A 


118  NEW  ADVANCED  ARITHMETIC. 

10.  Bought  a  farm  of  3244  acres  of  land  for  $15,646.  If 
the  house  and  a  house-lot  of  12  acres  were  counted  at  $3,150, 
what  price  per  acre  was  paid  for  the  rest  ? 

11.  What  number  is  that  }  of  which  exceeds  ^  of  it  by 
112? 

12.  What  number  is  that  §  of  j  of  which  is  291  less  than 
I  of  I  of  it? 

Owning  ^j  of  a  farm,  I  sold  ^  of  my  share.  If  what 
I  had  left  was  worth  $211),  what  would  the  whole  farm  be 
worth  at  the  same  rate? 

14.  I  bought  a  house  and  lot  for  $5,784,.  paying  j^^  as 
much  for  the  lot  as  for  the  house.  How  much  did  I  pay 
for  each? 

15.  I  of  A's  farm  equals  f  of  B's.  Together  they  have 
551  acres.     How  many  has  eacliT 

16.  A  and  B  can  do  a  certain  piece  of  work  in  12|  days. 
They  worked  togeMier  6|  days ;  A  then  left,  and  B  finished 
the  work  in  8J  days.  In  how  many  days  could  each  do  the 
work  alone? 

17.  How  many  bushels  of  grain  can  be  bought  for  $l,260f 
at  31^  cents  a  bushel?  at  56^?  at  66f  ?  at  87i? 

18.  If  56|  acres  of  land  cost  $1,200,  what  is  the  price  per 
acre? 

19.  How  long  will  225  pounds  of  meat  last  a  crew  of  5  men, 
at  the  rate  of  If  pounds  per  day  for  each  man? 

20.  A  man's  crop  of  oats  weighed  66,677  pounds.  He  sold 
it  for  3^  cents  a  bushel.  Counting  32  pounds  for  a  bushel, 
what  was  the  value  of  his  crop? 

21.  A  crop  of  wheat  weighing  90,144  pounds  brought 
$1,252.  Counting  60  pounds  to  a  bushel,  what  was  the  price 
per  bushel  for  wheat  sold? 

22.  f  +  A  +  i^rr  -  t  of  ^^^  =  ? 


FRA  CTIONS.  119 

23.  Bought  8J  yards  of  broadcloth  at  $4^  a  yard,  5J 
yards  of  cassimere  at  $2h  a  yard,  5?  yards  of  silk  at  $2^ 
a  yard,  and  paid  for  all  with  corn  at  41|  cents  a  bushel. 
How  many  bushels  were  required  ? 

24.  What  is  the  1.  c.  m.  of  824,  936,  1020? 

25.  A  man  owned  f  of  a  factory.  He  sold  |  of  his  share. 
He  gave  ^  of  the  remainder  to  his  daughter,  ^  of  what  then 
remained  to  hi^  son,  and  sold  J  of  the  remainder  for  §14,000. 
What  was  the  value  of  the  factory?  What  was  the  daugh- 
ters share  ?  the  son's  share  ?  What  was  the  value  of  what 
he  had  left  ? 

26.  Find  the  sum,  difference,  and  product  of  4£  and  6-j\. 

27  Find  the  quotient  arising  from  dividing  the  sum  of 
8/3  and  5f  by  their  difference. 

28.  A  can  do  a  piece  of  work  in  15  days,  B  in  12  days, 
and  C  in  10  days.  A  works  2  days,  B  3  days,  and  C  3 
days.  In  what  time  can  A  and  B  finish  the  job  by  working 
together? 

29.  Thomas  can  dig  a  ditch  in  3|  days,  Richard  in  6f 
days,  Harry  in  4*  days.  In  what  time  can  all  complete  it 
working  together? 

30.  Divide  f  of  6§  by  §  of  y^  of  12|. 

31.  X  f  X  -f  X  16|  X  ^0. 

32.  A  cistern  having  a  capacity  of  88|^  barrels  contained 
63/o  barrels.  After  15^^  barrels  were  pumped  out,  the  cis- 
tern was  what  part  full  ? 

33.  A  had  a  journey  of  43^2  miles  to  perform.  After 
traveling  36^  miles,  what  part  of  the  journey  remained  ? 

34.  23  is  what  part  of  48i?  of  614?  of  58§? 

35.  T^  is  what  part  of  87?  of  169?  of  4:6^?  of  83tV? 

36.  7|  is  what  part  of  lof?  of  28|?  of  72|? 


120  NEW  ADVANCED  ARITHMETIC. 

37.  Divide  76  into  two  such  parts  that  f  of  the  first  shall 
equal  ^  of  the  second. 

38.  Divide  417  into  three  such  parts  that  the  first  shall 
be  j\  of  the  second,  and  the  third  shall  be  \^  of  the  second. 

39.  How  is  the  value  of  a  proper  fraction  affected  by 
adding  the  same  number  to  both  terms?     Why? 

40.  How  is  the  value  of  an  improper  fraction  affected  by 
adding  the  same  number  to  both  terms?     Why? 

41.  How  is  the  value  of  an  improper  fraction  affected  by 
subtracting  the  same  number  from  both  terms?  Why?  Of 
a  proper  fraction  ?     Wh}-? 

42.  What  is  the  circumference  of  a  wheel  that  makes  24 1 
revolutions  in  400  feet? 

43.  If  another  wheel  have  a  tire  two  feet  shorter,  it 
makes  how  many  more  revolutions  in  the  same  distance  ? 

44.  It  takes  six  hours  to  complete  a  certain  journey  at 
the  ordinary  rate  of  travel.  How  many  hours  are  required 
to  complete  three  fourths  of  the  journey  if  the  rate  be  in- 
creased by  one  half  of  itself? 

45.  An  adult  inspires  30  cubic  inches  of  air  at  an  ordinary 
inspiration.  If  they  breathe  1<S  times  per  minute,  how  long 
■will  it  take  50  adults  to  breathe  once  all  the  air  in  your 
school-room  ? 

46.  If  each  expiration  vitiates  a  cubic  foot  of  air,  how 
rapidly  should  the  air  be  changed  in  the  above  room? 

47.  How  many  lunar  months  of  29i  days  in  the  year  of 
365^  days? 

Analysis.  There  are  as  many  lunar  months  of  29^  days  in  365 J  days, 
as  .365^  is  times  29^.  365 J  is  12^j\  times  29^  ;  hence  tiiere  are  12/]5j  lunar 
months  in  the  year. 

Note.  When  the  divisor  is  a  mixed  number,  it  is  usually  best  to  mul- 
tiply both  dividend  and  divisor  by  the  denominator  of  the  divisor.  In  the 
fores^oing  problem,  multiplying  both  dividend  and  divisor  by  4,  we  have 
1461  ^  118. 


FRA  CTTONS.  121 

48.  What  is  the  daily  journey  of  the  earth  in  its  orbit, 
292,000,000  miles,  allowing  365^  days  for  a  year? 

49.  Allowing  1^  cubic  feet  to  the  bushel,  what  is  the  capa- 
city of  a  bin  6'   X  8'   X  10'. 

50.  A  bushel  of  corn  in  the  ear  occupies  2^  cubic  feet. 
To  what  height  must  a  rail-crib  nine  feet  square  be  filled  to 
hold  300  bushels  ? 

51.  Measure  a  wagon-box  and  calculate  to  what  height  30 
bushels  of  corn  in  the  ear  will  fill  it;  40  bushels  of  wheat. 

52.  A  man  bought  a  piece  of  property  for  $1,050  and  sold 
it  so  as  to  gain  tt^  of  the  cost.    What  did  he  receive  for  it  ? 

53.  By  selling  a  piece  of  property  for  $4,820,  the  seller 

331 
gained  rr^  of  the  cost.     What  did  it  cost  him  ? 

371 

54.  Sold  a  horse  for  $75,  losing  tttt^  of  the  cost.    What 

was  the  cost? 

55.  The  sign  %  is  used  instead  of  the  word  "  hundredths" 
or  the  denominator  100;  read  it  in  that  way.  Find  25%  of 
80;  of  92;  of  144 ;  of  420;  of  610;  of  712. 

56.  Find  75%  of  64;  of  72;  of  112;  of  320;  of  684;  of 
860;  of  1024. 

57.  Find37i%  of  40;  of  112;  of  280 ;  of  576;  of  1320; 
of  1648;  of  1864. 

58.  Bought  goods  for  $60  and  sold  them  for  $75.  The 
gain  is  how  many  %  of  the  cost? 

59.  75%  of  375  is  2^%  of  what  number? 

60.  Find  the  1.  c.  m.  of  24,  36,  40,  48,  60. 

61.  Find  the  sum  of  §,  |,  |,  |. 


122  NEW  ADVANCED  APdTHMETIC. 

62.  Divide  $6,600  among  4  persons  so  that  the  second 
shall  receive  twice  as  much  as  the  first,  the  third  three  times 
as  much  as  the  second,  and  the  fourth  four  times  as  much  as 
the  third.  ^ 

63.  (31  +  2^)  X  (4i  -  If)  -  I  of  ^%  =? 

64.  2i  is  what  part  of  6/5  ?  of  8f  ?  of  9^  ?  of  lOf  ? 

H 

65.  §  is  j%  of  what?  Tf  is  f g  of  what?   ~  is  /^  of  what? 

66.  Divide  |  of  |  by  |  of  11. 

67.  Divide  f|  of  7 J  by  f±  of  fg  of  ||. 

68.  Add  17i,  26|,  69|,  142H,  SITH- 

69.  Add  242,  63|,  96U,  38|,  124^^. 

70.  64/,  -  42H  ;  69i|  +  26U  ;  315^  -  278^. 

71.  84?x63|;  275t«5X49|;   Y^^^ 

72.  m-^U  64H--5;  65^-^6;   1261* -- 7. 

73.  An  iceman  delivered  a  block  of  ice  1  foot  long,  §  of  a 
foot  wide,  and  ^  of  a  foot  thick,  and  charged  for  50  pounds. 
What  was  the  shortage  in  weight,  a  cubic  foot  of  water 
"weighing  Q2h  pounds,  ice  weighing  -^^j^  as  much? 

74.  A  steel  beam  is  16  feet  long,  2*  inches  thick,  and  14 
inches  wide.  "What  is  its  weight,  its  specific  gravity  being 
7.84  (7.84  as  heavy  as  an  equal  volume  of  water)  ? 

75.  What  is  the  weight  of  a  block  of  limestone  Qh  feet  X 
11  ft.  X  43  feet,  its  specific  gravity  being  2.62? 

76.  How  many  pounds  of  7,000  grains  will  8100,000  in 
ten-dollar  gold-pieces  weigh,  each  piece  weighing  258  grains? 

77.  If  A  can  do  a  piece  of  work  in  9  days,  B  in  10  days, 
and  C  in  12  days,  in  what  time  can  they  do  it,  working 
together?  If  they  should  work  together  1  day,  and  C  should 
leave,  in  what  time  could  A  and  B  finish  it?  If  they  should 
work  together  2  days,  and  B  and  C  should  leave,  in  what 
time  could  A  finish  it? 


FRACTIONS.  123 

78.  Make  a  receipted  bill  for  the  following  items,  using 
tiie  name  of  a  class-mate  with  your  own :  — 

24^  lbs.  sugar  at  6^  cts. 
18 'lbs.  coffee  at  27 J  cts. 
15  bu.  potatoes  at  36^  cts. 
15^  lbs.  butter  at  25  cts. 

79.  Make  a  bill  of  the  following  items :  A.  R.  Brown 
bought  of  G.  G.  Johnson  8  gallons  oil  at  12^  cents,  14 
brooms  at  20  cents,  1  pkg.  gold-dust  25  cents,  3  cakes  soap 
at  6  J  cents,  5  gallons  gasoline  at  10  cents,  16  pounds  sugar 
at  6:^  cents,  300  lemons  at  1^  cents,  2  pine-apples  at  35 
cents.  Credit  the  bill  with  $5.60  paid  on  account  and  show 
the  balance  due. 

80.  Make  a  bill  of  the  following:  P.  A.  Coen  &  Son  sold 
to  Augustine  &  Co.,  February  20,  1  quart  mucilage  50  cents, 

1  jar  paste  20  cents;  March  2,  1  quart  ink  50  cents;  (13)  10 
reams  cap  at  $1.35  ;  April  6,  30  pounds  paper  at  7  cents ;  (9) 

2  quarts  ink  at  50  cents ;  (13)  24  quinine  bottles  at  4^  cents ; 
(17)  90  pen-wipers  at  If  cents,  1  floor  brush,  $3.00;  May  6, 
2  gallons  benzine  at  20  cents;  (12)  31  pounds  paper  at  7 
cents,  5  reams  cap  at  $1.35;  (14)  5  gallons  turpentine  at  50 
cents;   (16)  5  reams  cap  at  $1.35. 

81.  If  $80  be  paid  for  the  labor  of  12  men  for  5  days,  at 
the  same  rate  what  should  be  paid  for  the  labor  of  18  men 
for  24  days  ? 

82.  If  $98  be  paid  for  the  use  of  $560  for  2  years  and  6 
months,  at  the  same  rate  what  should  be  paid  for  the  use  of 
$825  for  3  years  and  9  months? 

83.  At  the  rate  of  6J  cents  for  the  use  of  $1  for  a  year, 
what  should  be  paid  for  the  use  of  $650  for  3  years?  for 
$875?  for  $1,280? 

84.  At  the  same  rate  what  should  be  paid  for  the  use  of 
$1,460  for  3  years  and  4  months?  for  2  years  and  5  months? 
for  4  years  and  1 0  months  ? 


124  NEW  ADVANCED  ARITHMETIC. 

85.  How  many  bushels  of  corn  can  be  bought  for  $1,936.40 
at  23^  cents  a  bushel  'i 

86.  Sold  1,836  bushels  of  wheat  at  61 1  cents  and  invested 
the  proceeds  in  corn  at  24j|  cents ;  how  many  bushels  did  1 
buy? 

87  Built  a  fence  around  a  square  lot  400  feet  on  a  side. 
The  posts  were  placed  8  feet  apart.  The  boards  were  16 
feet  long  and  each  contained  8  feet  of  lumber.  If  the  fence 
was  5  boards  high,  what  was  the  cost  o*"  ♦^^he  material  (not 
counting  the  nails),  if  the  posts  cost  18  cents  each  and  the 
lumber  cost  Si 8. 50  for  1,000  feet? 

88.  What  is  the  cost  of  the  lumber  for  flooring  and  ceiling 
a  room  24  feet  and  3  inches  wide  and  84  feet  long,  if  the 
lumber  cost  3^  cents  a  square  foot? 

89.  A  field  is  40  rods  (IQ^  feet)  wide  and  80  rods  long. 
What  will  it  cost  to  plow  the  field  at  02.25  an  acre  (160 
square  rods)  ? 

90.  The  diameter  of  the  earth  is  about  8,000  miles.  Mt. 
P'.verest  is  about  oh  miles  high.  On  a  globe  2  feet  in 
diameter,  how  high  should  a  projection  be  to  represent  this 
mountain's  height  properly? 

91.  If  a  continent  6,000  miles  long  be  represented  by  a 
raised  map  4  feet  long,  what  horizontal  distance  does  a  foot 
represent?  an  inch?  How  high,  according  to  the  same  scale, 
should  Mt.  Everest  be? 

92.  How  many  leaves  in  Webster's  International  Diction- 
ary? The  thickness  of  one  leaf  compares  how  with  the 
thickness  of  the  book,  not  counting  the  covers?  Compare 
this  result  with  that  obtained  in  Problem  'JO. 

93.  If  7  cents  be  paid  for  the  use  of  $1  for  one  year, 
what  should  be  paid  for  the  use  of  $2,460  for  4  years,  7 
months,  15  days?     (Call  15  days  ^  of  a  month.) 


DECIMAL  FRACTIONS.  125 


169.     DECIMAL  FRACTIONS. 

1.  Define  fraction,  numerator,  denominator,  common 
fraction. 

2.  A  decimal  fraction  is  a  fraction  whose  denominator  is 
expressed  by  tlie  position  of  ttie  right-liand  figure  of  tlie 
numerator  witli  respect  to  the  decimal  point. 

3.  Since  we  employ  a  decimal  system  of  notation,  the 
denominator  of  a  decimal  fraction  is  a  power  of  ten. 

4    Decimal  fractions  are  commonly  called  "decimals." 

5.  A  decimal  whose  numerator  is  an  integer  is  a  pure 
decimal. 

6.  A  decimal  whose  numer  iior  contains  a  fraction  is  a 
complex  decimal. 

7.  An  integer  plus  a  decimal  is  called  a  mixed  decimal. 
8     Decimals  may  be  both  mixed  and  complex. 

9.  Pure  decimals:   .3,  .027,  .346,  .0001. 

10.  Complex  decimals:  .3^,  .0^,  .756f. 

11.  Mixed  decimals:  3.2,  7.001,  300.010. 

12.  Mixed  complex  decimals:  7.003,  562.0^,  27. 3§. 

170.    Reading  of  Decimal  Fractious. 

1.  Use  "and"  only  in  reading  mixed  decimals  or  mixed 
numbers. 

Note  also  that  such  forms  as  .\,  .%  are  never  used. 

2.  300.026  =  three  hundred  and  twenty-six  thousandths. 

3.  .326  =  three  hundred  twentj'-six  thousandths, 

4.  .0^  =  one-third  of  a  tenth. 

5.  7.00f  =  seven  and  three  fourths  of  a  hundredth. 

6.  .07^  =  seven  and  three  fourths  hundredths. 

7.  3.2  =  three  and  two  tenths.     (Mixed  decimal.) 

8.  3.2  ~  thirty-two  tenths.     (Pure  decimal.) 


126 


NEW  ADVANCED  ARITHMETIC. 


171.    Names  of   the  orders  to  the  right   of   the   decima] 
point : 

First,  tenths'  order. 

Second,  hundredths'  order. 

Third,  thousandths'  order. 

Fourth,  ten-thousandths'  order. 

Fifth,  hundred-thousandths'  order. 

Sixth,  millionths'  order. 

Seventh,  ten-millionths'  ordei'. 

Eighth,  hundred-millionths'  order. 

Ninth,  billionths'  order. 

Tenth,  ten-billionths'  order. 

Eleventh,  hundred-billionths'  order. 

Twelfth,  trillionths'  order. 

172.     NUMERATION. 
Read  the  following  numbers  : 

1.  .5,  .05,  .004,  .0006. 

2.  .00008,  .000009, .0000009. 

3.  .00000002,  .000000003. 

4.  .000000007,  .0000000001. 

5.  .26,  .264,  .3864,  .029. 

6.  .0294, .00874,  .087463. 

7.  .0056849,  .00046928. 

8.  .000057006,  .600870543. 

9.  .0080500694,  .4060790843. 

10.  .00400014,  .100010014. 

11.  .2002000202,  .33033003. 

12.  7.003,  .07f,  .0300,  .310,  300.010. 

13.  .Of,  .00^,  .OOOf,  15.0^,  29.00^. 

RULE. 
Read  the  nutnber  as  if  it  were  integral,  and  then  apply 
the  denomination   indicated    by  the  position  of  the   right- 
hand  figure  of  the  nunieratorm 


A 


DECIMAL   FRACTIONS.  127 

Read  mixed  decimals,  first,  as  mixed  numbers,  and  second, 
as  simple  nmnbei's. 

Illustration.    568.00861. 

As  a  mixed  number,  568  and  861  hundred-thousandths. 

As  a  simple  number,  56  million  800  thousand  861  hundred- 
thousandths. 

In  reading,  make  a  slight  pause  before  giving  the  denomi- 
nation.    Read  the  following: 

1.  29.00863,  4609.001083. 

2.  200183.40062,  69.00185706. 

3.  30086.0030086,  4000.004. 

4.  5000000.000005,  .008 J. 

5.  .0569 1-1^,  .0003,  .00002-. 

6.  .000006i-i,  -0700,  17.04200. 

7.  820.007600,  .4500,  .07900. 

173.     NOTATION. 

1.  There  will  be  little  facility  in  writing  decimal  fractions 
until  pupils  are  thoroughly  familiar  (a)  with  the  names  of 
the  orders,  and  (b)  with  the  number  of  each  order,  counting 
from  the  decimal  point  toward  the  right. 

2.  What  is  the  number  of  the  following  orders:  hun- 
dredths'? millionths'?  tenths'?  ten-thousandths'?  teu- 
milliouths'  ?  hundred-thousandths'  ?  hundred-million  ths'  ? 
billionths'  ? 

3.  Name  each  of  the  following  orders :  4th,  8th,  1st,  2d, 
7th,  5th,  3d,  9th,  6th,  10th. 

4.  Steps  in  writing  decimal  fractions  : 

(1)  Thinli  the  number  of  the  order  in  which  the  right-hand 
figure  of  the  numerator  must  stand. 

(2)  Think  the  number  of  figures  in  the  niunerator. 

(S)  The  difference  between  these  numbers  is  the  number 
of  ciphers  between  the  numerator  and  the  decimal  point. 


128  NEW  ADVANCED  ARITHMETIC. 

174.  Illustration.     Write  in  figxires  469  ten-millionths. 

1.  In  order  that  a  number  shall  express  ten-millionths,  its 
right-hand  figure  must  stand  in  the  seventh  order  to  the  right 
of  the  decimal  point. 

2.  In  469  there  are  three  figui-es ;  hence, 

3.  Four  ciphers  must  precede  the  numerator. 

To  express  this  number,  consequently,  I  write :  decimal 
point,  four  ciphers,  469. 

175.  Tell  how  each  of  the  following  numbers  is  expressed 
before  you  make  any  figures. 

1.  Write  in  figures  17  hundred- thousandths. 

Illustration.  This  number  is  expressed  by  writing  decimal  point,  three 
ciphers,  one,  and  seven. 

2.  Write  in  figures  6  tenths;  15  hundredths;  43  thou- 
sandths ;  28  ten-thousandths ;  467  thousandths. 

3.  Write  in  figures  29  hundredths  ;  921  thousandths ;  7 
ten-thousandths ;  8  hundred-thousandths ;  4  millionths ;  6 
billionths ;  38  ten-thousandths  ;  4562  hundred-thousandths. 

4.  419  millionths;  306  hundred-thousandths;  96^  ten- 
thousandths;  8158  hundred-milliouths;  59001  billionths. 

5.  23006  and  40007  millionths;  29000  and  29  thou- 
sandths; 29029  thousandths;  307  million  and  307  mil- 
lionths;   307000307  millionths. 

6.  379  tenths;  5824  hundredths;  69708  thousandths; 
524896  hundred-thousandths. 

7.  Three  million  seventeen  thousand  eight  hundred- 
billionths. 

8.  Seven  hundred  ten-thousandths;  seven  hundred  ten 
thousandths ;  nine  thousand  two  hundred-millionths ;  nine 
thousand  two  hundred  millionths.      ^ 

9.  Two  thirds  of  a  millionth ;  five  sixteenths  of  a  ten- 
thousandth ;  eighty-nine  and  seven  thirteenths  hundred- 
millionths- 


DECIMAL  FRACTIONS.  129 

10.  Eight  aud  eight  thousandths. 

11.  Twenty-three  aud  sixty-one  ten-thousandths. 

176.     REDUCTION   OP   DECIMAL    FRACTIONS. 

1.  Reduce  8  to  tenths;  24  to  thousandths;  .5  to  hun- 
dredths; .75  to  ten-thousandths;  .0624  to  millionths. 

2.  Reduce  .2600  to  hundredths;  .050800  to  ten-thou- 
sandths; 18.000  to  hundredths;  to  tenths;  75.0630000  to 
millionths ;  to  thousandths. 

3.  Reduce  4. 6  to  hundred-thousandths ;  reduce  the  result- 
ing fraction  to  hundredths;  87.2  to  billionths ;  the  resulting 
fraction  to  millionths  ;  to  tenths. 

4.  Annexing  a  cipher  to  the  numerator  of  a  decimal  mul- 
tiplies the  numerator  by  ten.  But  in  making  one  more  place 
in  the  numerator,  it  multiplies  the  denominator  by  ten, 
hence : 

Annexing  ciphers  to  the  right  of  a  decunal  does  not 
change  its  value. 

177.     REDUCTION    OF   DECIMAL   FRACTIONS    TO 
COMMON   FRACTIONS. 

"\ATiat  is  the  difference  between  a  decimal  fraction  and  a 
common  fraction?  What  must  be  done,  then,  to  change  a 
decimal  fraction  to  a  common  fraction?  Change  the  follow- 
ing to  common  fractions. 

1.  .5.         4.    .000637.         7.    3.08.  10.    .00^. 

2.  .37.       5.    .004826.         8.    25.0063.       11.    .00285yV 

3.  -085.     6.    .04725.  9.    .0056^.         12.    18.0675/t. 

RULE. 
To  change  a  decimal    fraction  to  a  common  fraction: 
Erase    the    decimal    jfoint,    tvrite    the   denominator,    and 
reduce  the  fraction  to  longest  terms. 

What  will  the  denominator  always  Ije  before  reduction? 


130  NEW  ADVANCED  ARITHMETIC. 


1. 

.8J, 

.08^,     .( 

)08^. 

2. 

1.62, 

.16^, 

.0162,     .00016|. 

3. 

.333S 

.0331, 

.311,     .0311. 

4. 

.431, 

.00433, 

.581,     .0058J. 

5. 

.56V, 

.0681, 

.00661,     .0811. 

6. 

•Oi-, 

.0002, 

.01,     .OOiV- 

7. 

•0|, 

•OOA, 

.000-iV»     -000^. 

179.     lUastrath-e  Problem. 

1.  Change  ^^  to  a  decimal  fraction.  Tiiis  fraction  is  to  be 
regarded  as  a  problem  in  partition,  iu  which  the  numerator 
is  the  dividend  and  the  denominator  is  the  divisor.  |  of  1 
equals  ^  of  3. 

Analysis  Since  |  of  3  is  not  a  whole  number,  3  is  reduced  to  tenths. 
3  =  3.0.  \  of  3.0  =  .3  with  a  remainder  of  .6.  .0  =  .60.  ^  of  .60  =  .07 
with  a  remainder  of  .04.     .04  =  .040.     \  of  .040  =  .005  ;  hence  |  =  .375. 

FORM. 

^  — The  reductions  may  be  made  at  once  as  above. 

.375 

2.  How  did  we  show  that  the  3  was  reduced  ?  Explain 
the  following 

RULE. 

To  reduce  a  conitnon  fraction  to  a  ttecitnal :  Place  the 
decimal  point  at  the  right  of  the  numerator,  annc.r  zeros 
and  divide  by  the  denominctor.  Point  off  as  many  deci' 
mats  in  the  quotient  as  there  icere  zeros  annexed. 

180.      PROBLEMS. 
Change  the  following  to  decimal  fractions. 

•      4'         if         tl         ^1         1?»         ^1  8   • 

2-    VVi     t'V'      tV'     t's'i      inri     ^T)i     ^• 

3.  ^si     ii;     M'     ^Vi     -svii     ih     ■g'jy 


DECIMAL  FRACTIONS.  131 

4.  M.     3^'     Ih     t%     h^     /^^5,     its- 

5.  io*        'fzi        ^WOi        4  0%'        4  0U5        ^5(J»        T25^* 

Note.  The  foregoiug  fractions  all  reduce  to  pure  decimals  because 
each  reduced  numerator  is  divisible  by  its  denominator.  A  study  of  the 
denominators  shows  that  the  only  factors  of  the  denominators  not  found 
in  the  corresponding  numerators  are  2  and  5.  But  these  are  the  factors 
that  are  introduced  into  the  numerator  with  each  reduction;  hence  it  is 
possible  to  carry  the  reduction  far  enough  to  make  each  numerator 
divisible  by  its  denominator. 

Note.     Dictate  many  problems  until  this  becomes  plain. 

6.      35     75     ^J1     1^55      24'     45* 

Note.  If  the  denominator  of  a  fraction  in  lowest  terms  contains  other 
factors  than  2  or  5,  it  cannot  be  reduced  to  a  pure  decimal.  In  such  cases 
it  is  found,  as  the  division  continues,  that  a  certain  figure  or  set  of  figures 
is  regularly  repeated.  This  figure,  or  set  of  figures,  is  called  the  repetend, 
and  "he  decimal  in  which  it  occurs  is  called  a  repeating,  or  circulating 
decimal. 

I  =  .33333333.333,  etc. 

^  =  .636363636363,  etc. 

^  =  .428571428.571428571428571,  etc. 

In  .«uch  cases  we  may  express  the  common  fraction  as : 

1.  A  complex  decimal,  .33^,  .&S^^,  or  .6y\,  .428f 

2.  An  approximate  decimal,  .3333+,  .6363+,  .4286-.  The  sign  +  or  — 
is  used  to  show  that  tlie  result  is  incomplete,  and  that  the  true  value  is 
greater  or  less  than  the  value  expressed. 

3.  A  repeating  decimal,  .3,  63,  .428571  — .  Dots  are  placed  over  the 
terminal  figures  of  a  repetend. 

7.  Reduce  the  following  to  eacn  of  the  above-named 
forms,  making  the  approximate  decimals  true  to  ten- 
%ousandths. 

t\'  f'  ih  +4.  vh  ^?'  M'  if'  §§• 

181.    Another  Method. 

Illustrative  Problem.     Change  |  to  a  decimal  fraction. 

Analysis.  Since  the  denominator  of  a  decimal  fraction  is  a  power  of 
ten,  I  multiply  both  terms  of  |  by  some  number  that  will  make  the  denom- 
inator a  pov.er  of  ten.     Every  power  of  ten  is  the  product  of  an  equal 


132  NEW  ADVANCED  ARITHMETIC. 

number  of  twos  aud  fives.  P^iglit  is  the  product  of  three  twos.  If  it  be 
multiplied  by  the  product  of  three  fives,  the  result  will  be  the  product  of 
three  tens,  which  is  tiie  third  power  of  ten.  Multiplying  both  terms  of  J 
by  125,  the  result  is  jVwj  or  .875. 

Explain  the  following  by  this  method. 
Keduce  to  decimal  fractions  : 

1  1       2.       -J-       ja.       1.5        21       41 
■•■•      25      55     20?     255     T  l>  5     325     tii' 

2  -^  7  1  1         _1_  1  7  5  a 

8^'  12^'  16f'  6i'  33i'  6§'  2>\ 

RULE. 

Multiply  or  divide  both  terms  of  the  fraction  frti  some 
number  that  uill  make  tlie  denominator  a  itower  of  ten. 
Then  ea-press  the  denominator  by  the  position  of  the 
riyht-hand  figure  of  the  numerator  with  respect  to  the 
decimal  point* 

182.    ADDITION   OF   DECIMAL   FRACTIONS. 

Define  addition,  sum.  Numbers  are  "written  how? 
Why?  Addition  begins  where?  Why?  Make  a  rule. 
Analyze  each  problem  as  in  simple  addition. 

183.     PROBLEMS. 

1.  2                           3                               4. 

.625  4.073     126.0009     4006.092 

.0984  26.0084     482.1872        9.059701 

.4907  59.00462    600.50983     683.086409 

.00864  83.01879     17  008459      34.189357 

.09769  50.00043      6.098725    1086.049078 

\ 

5.  Add  .00862,  4.04378,  73.096,  168.00097,  49.287005, 
83.460037. 

6.  Add  77.02081,  94.09069,  88.00799,  686.060098, 
897.0609,  .084858,    .087857,    .3060686,    .76978. 

7.  Add  .04069,  .008972,  .0934,  .0083462,  .027309, 
.5302681,  .05003701. 


DECIMAL  FRACTIONS.  133 

8.  Add  7  tenths,  56  hundredths,  93  thousandths,  329 
hundred-thousandths,  8052  millionths,  42067  ten-thou- 
sandths,   43   hundredths,    98   ten-millionths. 

184.    ADDITION   OF   COMPLEX   DECIMALS. 

Illustrative  Example.  Add  3^  tenths,  7f  thousandths, 
17-i2r  hundredths,  86 2|  ten-thousandths. 

Analysis.  Ten-thousandths  being  the  lowest  denomination,  all  num- 
bers of  higher  denomination  are  to  be  reduced  to  teu-thousaudths,  unless 
they  become  pure  decimals  before  such  reduction  is  completed. 

.3^        =      .3  +  .^i-^  =  .3125 

.007f    =  .007  +  .^5?a  =  .0074 

.17t\    =    .17  +  .-4f^  =  .17l8fV 

.0862|  =  .08621 

Note.  If  all  of  the  mixed  decimals  can  be  reduced  to  pure  decimals, 
reduce  them  before  addition. 

1.  Add  3J  tenths,  29^  thousandths,  56]  thousandths, 
24^  hundredths,   183^  thousandths,  86^  hundredths. 

2.  Add  15f  thousandths,  38^  hundredths,  409  .\  ten- 
thousandths,  3^  tenths,  9^  hundredths,  7f  thousands,  5-^j 
tenths,  2^  hundredths. 

3.  Add  64|  hundred- thousandths,  53*  hundredths,  78.^ 
thousandths,  86|  ten-thousandths,  4920/^  hundred-thou- 
sandths, 6/2  tenths,  9/5  hundredths. 

4.  Add  423W  millionths,  29^^  hundredths,  46J  units, 
126§  tenths,  479^V  hundredths. 

185.    SUBTRACTION   OP  DECIMAL   FRACTIONS. 

Define  subtraction,  minuend,  subtrahend,  remainder. 
Perform  and  explain  a  problem  in  subtraction  of  simple 
numbers.  Give  the  rule.  Apply  the  same  to  the  fol- 
lowing problems. 

Note.  Make  minuend  and  subtrahend  of  the  same  denomination  be>- 
fore  subtracting. 

lOA 


134  NEW  ADVANCED  ARITHMETIC. 

186.  PROBLEMS. 

1.  .0861  -  .0295  =  ? 

2.  .7043  —  .4805  =  ? 

3.  .0461  -  .00356  =  ? 

4.  4.02603  -  .9078  =  ? 

5.  26.1059  -  19.74308  =  ? 

6.  461.083024  —  86.59260834  =  ? 

7.  .023  —  .000465  =  ? 

8.  92.  —  .06479  =  ? 

9.  400.  —  .00004  =  ? 

10.  .00583  -  .0000583  =  ? 

11.  .7^  -  .0081  ^  ? 

12.  .43J  —  .0047^  =  ? 

13.  .0084}  —  .  00023 1^^-  =  ? 
Note.     .0084}  =  .00841^  =  .00840^. 

14.  42.08^  -  34.0574H  =  ? 

15.  83.50703xV  —  59.02  =  ? 

16.  .80352  _  .0454  =  ? 

17.  84.0431  +  56.057  —  16.059  -  23.00845  =  ? 

18.  625  —  .0748  +  29.0536  -  439.00596  -  .089037  =  ? 

19.  .08^  +  .0043^  —  .0061  =  ? 

20.  5^  —  .061  +  .009^^  +  -OO-tV  =' 

21.  28  thousandths  —  46  ten-thousandths  =? 

22.  423  millionths  —  17  hundred-thousandths  =? 

23.  46  tenths  —  46  thousandths  =? 

24.  3824  hundredths  —  3824  ten-millionths  =? 

25.  \  of  a  tenth  —  |  of  a  thousandth  =? 

26.  ^  of  a  thousandth  —  ^  of  a  millionth  —  ? 

27.  426|  ten-thousandths  —  38^  hundred-thousandths  =? 

28.  9251  hundredths  —  4659  hundred-millionths  —? 


DE  CI  MA  L  FRAC  TIOXS.  135 

29.  94f  thousandths  —  53}|  ten-millionths  =? 

30.  \  of  ten  —  ]  of  a  hundredth  =? 

31.  ^  of  one  hundred  —  ^  of  a  hundredth  =? 

187.  MULTIPLICATION  OF  DECIMAL  FRACTIONS. 

1.  Define  all  terms  in  multiplication. 

2.  "What  is  the  effect  of  removing  the  decimal  point  one 
place  to  the  right?  t'^\o  places?  four  places?  How  multiply 
by  10?  by  100?  by  10000?  by  1000000? 

3.  "What  is  the  effect  of  removing  the  decimal  point  one 
place  to  the  left?  three  places?  six  places?  How  divide  by 
10?  by  1000?   by  1000000? 

4.  Multiply  .008764  by  10,  and  read  the  result;  by  100; 
by  1000;  by'lOOOOOO;  by  10000000. 

5.  Divide  4968.307  by  10,  and  read  the  result;  by  100; 
by  1000;  by  10000. 

6.  Mate  general  rules  for  multiplying  and  dividing  by 
powers  of  ten. 

188.  1.  Multiply  .0536  by  28. 

Note.  Since  tlie  multiplicand  is  ten-tliousandtlis,  the  product  is  ten- 
thousandths;  hence,  28  times  .0536  =  1.5008. 

2.  Multiply  .824  by  .01 ;  by  .001 ;  by  .1 ;  by  .0001,  and 
read  the  results. 

Multiplying  a  number  by  .1  is  equivalent  to  dividing  it  by 
what  divisor? 

3.  Multiply  .497  by  .39.  This  problem  means:  Find  .39 
of  .497.  How  do  you  find  .01  of  .497?  What  will  the 
result  be?  How  many  decimal  places  will  it  have?  "What 
do  you  do  with  this  result? 

4.  Explanation.  I  first  find  .01  of  .497.  .01  of  .497  is 
.00497,  which  is  found  by  removing  the  decimal  point  two 
orders  to  the  left,  and  filling  the  vacant  orders  with  ciphers. 
.39  of  .497  is  39  times  .00497. 


136  NEW  ADVANCED  ARITHMETIC. 


(1)  (2) 

.497  }  S  .00497 


197  > 
.39  J 


39 


.04473 
.1491 


.19383 

5.    How  many  decimal  places  are  there  in  the  product? 
Why?     Make  a  rule  for  "pointing"  the  product. 

RULE. 
In  Multiplication  of  Decimal  Fractions,  multiply  as  in 
simple  numbers,  and,  point  off  as  many  decimal  places  in 
the  product  as  there  are  in  multiplicand  and  multiplier. 

189.      PROBLEMS. 
Multiply : 

1.  .542  by  58.        3.   .00436  by  .8.        5.   6.0281  by  .072. 

2.  .0693  by  324.    4.    .7093  by  .49.        6.   28.0563  by  .057^. 
Note.     Simplify  the  decimal  fractions. 

7.    .800694  by  .17^-  8.    .006294^  by  .00863f. 

9.  .0976^  by   24^^. 

Note.     Change  ^s  to  a  decimal  fraction. 

10.  25864  by  .03972. 

11.  30.6895  by  4.300906. 

12.  50638  thousandths  by  9026  hundredths. 

13.  49060037  millionths  by  207003  ten-thousandths. 

14.  409  billionths  by  36  millionths. 

15.  5063087  teu-millionths  by  6204  thousandths. 

16.  29  ten-thousandths  by  29  ten-millionths. 

17.  48  tenths  by  48  ten-thousandths. 

Give  the  results  rapidly  in  the  following  problems : 

18.  9  hundredths  by  7  thousandths. 

19.  15  ten-thousandths  by  6  thousandths. 

20.  8  thousands  by  9  thousandths. 


DECIMAL   FRACTIONS.  137 

21.  24  ten-thousands  by  24  ten-thousandths. 

22.  17  hundreds  by  4  hundred- thousandths. 

23.  23  tenths  by  4. 

24.  11  milliouths  by  11  thousandths. 

25.  18  ten-thousandths  by  4  hundredths. 

26.  7  ten-milliouths  by  14  hundred-thousandths. 

27.  What  is  the  product  of  tenths  and  thousandths?  of 
thousandths  and  hundredths?  of  ten-thousandths  and  ten- 
thousandths?  of  hundreds  and  thousandths?  of  thousands 
and  hundredths?  of  millions  and  millionths?  of  hundred- 
thousands  and  hundredths  ?  of  millionths  and  hundreds  ?  of 
ten-thousandthfe'.  and  thousands? 

28.  .125  is  what  part  of  1?  of  1  ten? 

29.  .375  is  what  part  of  1  ?  .0375  is  what  part  of  .1  ?  of  1  ? 

30.  .0025  is  what  part  of  .01  ? 

31.  .0625  is  what  part  of  .  1  ? 

32.  .000875  is  what  part  of  .001  ? 

33.  .006|  is  what  part  of  .01? 

Read  each  of  the  following  as  a  part  of  1  standing  in  the 
first  order  at  the  left  of  its  significant  figures : 

34.  .00075,  .0025,  .00125,  .0000875,  .033.^,  .0000662. 

35.  What  is  the  cost  of  864  bushels  of  oats  at  $0.41  per 
"bushel  ? 

36.  What  is  the  cost  of  17  horses  at  $112,375  each? 

37.  What  is  the  cost  of  384  acres  oi  land  at  $67,065  each? 

38.  What  is  the  cost  of  18.56  yards  of  cloth  at  $2.5625  a 
yard? 

39.  What  is  the  cost  of  29  books  at  $0.0625  each? 

40.  What  is  the  cost  of  465.375  bushels  of  wheat  at 
$0.9175  per  bushel? 


138  NEW  ADVANCED  ARITHMETIC. 

190.     DIVISION  OF  DECIMAL  FRACTIONS. 

1.  Define  partition,  dividend,  divisor,  quotient,  remainder. 
.97254-^8=:':'     Explain  by  partition.     Follow  the  form 

in  simple  numbers. 

2.  Explain  the  following  in  the  same  way. 

191.     The  divisor  a  whole  number. 

1.  .7658^7  =  ?  6.  .042864 -f- 24  =  ? 

2.  .04536  -^  6  =  ?  7.  .008399  -^  i}7  =  ? 

3.  .157032  -^  9  =  ?  8.  .010867  -^  46  =  ? 

4.  71.40636  ~  12  =  ?  9.  4.20638  4-  55  =  ? 

5.  146.0736  -^  16  =  ?  10.  436.0095  h-  77  =  ? 

11.  What  is  the  denomination  of  the  quotient  in  partition? 
In  the  preceding  problems  the  quotients  are  like  what?  Hcjw 
many  decimal  places  are  there  in  each  quotient? 

12.  Make  a  rule  for  pointing  the  quotient  when  the  divisor 
is  a  whole  number. 

13.  Explain  the  same  problems  by  division,  using  the 
form  given  in  simple  numbers. 

14.  The  quotient  in  each  case  is  like  what  ? 

RULE. 

In  dirision  of  decimal  fractions,  ulien  the  divisor  is  a 
whole  tiutnber,  divide  as  in  simple  nufiibers,  atid  point  off 
aa  niatiy  decimal  places  in  the  Quotient  as  there  are  in 
the   dividend, 

192.     The  divisor  a  decimal. 

1.    Illustrative  Problem.     .Sib  -f-  .5. 

The  dividend  is  the  product  of  the  divisor  and  quotient. 
The  number  of  decimal  places  in  the  product  equals  the  num- 
ber in  both  multiplicand  and  multiplier.  Hence  the  numl)er 
of  decimal  places  in  the  quotient  is  equal  to  the  number  iu 


DECIMA L   FRA  CTIONS. 


139 


the  dividend  minus  the  number  in  the  divisor.     In  this  case 
the  number  of  decimal  phices  in  the  quotient  is  3  —  1,  or  2. 

.5)  .875 
1775 
2.    87.5^.005  =  ? 

We  annex  ciphers  until  the  dividend  contains  as  many 
decimal  places  as  the  divisor.  The  quotient  contains  3—8 
decimal  place's. 

.005)  87.500 
17500 


193.       PROBLEMS. 


3.  .6241  -^  .79  11. 

4.  1.0276 -^  .028.  12. 

5.  44.814  -^  .;>7.  13. 

6.  .39071  -^  .0089.  14. 

7.  .091512  -~  .0124.  15. 

8.  7.5522  -^  2.46o  16. 

9.  .153032  ^    00376.  17. 

10.  .02336081  ^  .00583.         18. 


5.66703747  -^  70.83. 
.052629096  H-  5470.8. 
183.057  -h  .379. 
39.3888  -^  .0528. 
5.81715  -:-  .00695. 
.633447  -^  .000783. 
.7737013  -^  .0000859. 
112.1021  -^  .02453. 


How  does  the  rule  apply  to  this  problem  ? 


19 

20. 

21. 

22 

23 

24. 

25. 

33. 

34. 

35c 

36. 

37. 


2099.274  ^  .3607.  26. 

26624.32  -^  .4379.  27. 

5481  ^  ,063.  28. 

48760  ^  .0092.  29. 

766300  ^  .00079.  30. 

133574  —  .000329.  31. 

24980020  -^  .0406.  32. 


8.6947  -^  28. 
.46083  -^-  .37. 
.070685  ^ .000056. 


860.025 

90.3864 


.0378. 

4.77. 


.016458  -^  .000963. 
.023907  ^  .0001839. 


.7  -^  43,  carry  the  quotient  to  5  decimal  places. 

4.6-^58,         "  "  6       "  " 

63.  ~  97,         "  "  4       "  " 

,1  _i-  329  "  "  6       '  " 

4.  -^  586,         "  "  8  " 


140  NEW  ADVANCED   ARITHMETIC. 

194.  To  divide  by  a  power  of  10,  remove  the  decimal 
point  as  many  places  to  tlie  left  as  there  are  zeros  in  the 
divisor. 

1.  Divide  8493.7  by  10;  by  100;  by  10000;  by  1000000. 

2.  Divide  49.683  by  100;  by  10000;  by  1000;  by  10;  by 
1000000. 

195.  Division  by  the  factors  of  a  number. 

1.  Divide  8.75  by  50.    8.75  h- 10=  .875.    .875  ^5  =  .175. 

2.  Divide  12.25  by  500;  by  5000;  by  250;  by  25000. 

3.  Divide  75  by  750  ;  by  7500 ;  by  75000. 

4.  If  24  boxes  of  fruit  cost  $52.32,  what  does  each  box 
cost  ? 

196.      REVIEW    PROBLEMS. 

1.  Change  .00875  to  a  common  fraction. 

2.  Change  ^2  ^  ^  decimal  fraction. 

3.  How  many  square  feet  in  a  piece  of  ground  86.48  feet 
long  and  39.6  feet  wide? 

4.  If  a  man  travel  29.6  miles  a  day,  in  how  many  days 
will  he  travel  1,016.088  miles? 

5.  Change  ^^  to  a  decimal  fraction. 

6.  What  is  the  cost  of  473.5  bushels  of  corn  at  .47  of  a 
dollar  a  bushel? 

7.  At  .47  of  a  dollar  a  bushel,  how  many  bushels  of  corn 
can  be  bought  for  S222..545? 

8.  Add  342  thousandths,  568i  hundredths,  634^-  tenths, 
I  of  a  hundredth,  and  J  of  a  ten-thousandth. 

9.  From  }  of  a  tenth  take  ^  of  a  thousandth. 

10.  Change  .00^  to  a  common  fraction. 

11.  Change  .04  to  a  common  fraction. 

12.  Change  y^j  to  a  decimal  fraction  of  three  orders. 

13.  If  97  books  cost  $317,675,  what  will  each  cost? 


DECIMAL  FRACTIONS.  141 

14.  If  S8.5  bales  of  cloth  cost  $3,048.43,  what  is  the  price 
of  each  bale  ? 

15.  If  a  pair  of  shoes  cost  $2,625,  how  many  pairs  can  be 
bought  for  $49,875? 

16.  At  83  cents  a  yard,  how  many  yards  of  cloth  can  be 
purchased  for  $61,005  ? 

17.  At  $.045  per  pound,  how  many  pounds  of  sugar  can 
be  purchased  for  $1.89? 

18.  Define  a  decimal  fraction.  How  does  it  differ  from  a 
common  fraction  ? 

19.  Find  1.  c.  m.  of  18,  24,  36,  40,  180. 

20.  Define  a  pure  decimal ;  a  mixed  decimal. 

21.  23^  X  4f  =?  831  X  25H  =?  3^  X  24.08  =? 

22.  693|-f-7;  826i-^9;  938^^ -^  15. 

23.  Tell  how  to  change  a  decimal  fraction  to  a  common 
fraction.  Give  two  ways  of  changing  a  common  fraction  to 
a  decimal  fraction, 

H  X  4f     ^    .08;V  X  1     . 


24. 


—  ? 


25  What  common  fractions  can  be  changed  to  pure  deci- 
mals ?     Explain  in  two  ways  why  this  is  so. 

26.  2^  is  what  part  of  15?  of  20?  of  31?  of  6§?  of  8|? 
of 18|?  of  50?  of  100? 

27.  How  many  twos  are  there  as  factors  in  the  denomina- 
tor of  f  ?  How  many  successive  one-place  reductions  of  the 
numerator  must  be  made  to  introduce  these  twos?  How 
many  ciphers,  then,  should  be  annexed  at  once?  Same 
questions  for  ^^  ;  for  ^| ?     How  about  y^ ?  ^^? 

28.  Study  the  following  fractions  in  a  similar  way. 


fl  S  .39 

»         25'         T^5' 


50" 


JL9_ 


29.  In  addition  of  mixed  decimals,  what  should  be  done 
before  beginning  to  add?  Answer  a  similar  question  for 
subtraction. 


142  NEW  ADVANCED  ARITHMETIC. 

30.  Loaned  $824.40  with  the  uuderstanding  that  .06^  of 
the  amount  should  be  paid  me  for  its  use  for  one  year.  At 
this  rate,  what  sum  should  be  paid  for  its  use  for  18  months? 
for  30  months?  for  3  years  and  5  months? 

31.  AV^hat  is  the  rule  for  "  pointing"  the  product  in  multi- 
plication of  decimals?     Give  the  reason. 

32.  Sold  a  house  and  lot  for  $1,824.60,  losing  16^  per  cent 
of  the  cost;  what  was  the  cost? 

33.  What  shall  goods  costing  $624.80  be  sold  for  to  gain 
S~h  per  cent  of  their  cost? 

34.  Define  measurement,  and  all  terms.  Define  partition, 
and  all  terms. 

35.  Give  and  explain  the  rule  for  "pointing"  the  quo- 
tient when  the  divisor  is  a  decimal  fraction.  How  multiply 
by  a  power  of  10?     How  divide  by  a  power  of  10? 

36.  Multiply  324.086  by  2.5648.  Divide  831.2157728  by 
324.086. 

37.  There  are  2,150.42  cubic  inches  in  a  bushel.  What  is 
the  capacity  in  bushels  of  a  bin  48  feet  long,  8  feet  wide, 
and  12  feet  high? 

38.  There  are  231  cubic  inches  in  a  gallon.  What  is  the 
capacity  in  40-gallon  barrels  of  a  cubical  cistern  6x8  feet 
on  the  bottom  and  10  feet  deep? 

39.  A  brick  is  8  X  4  X  2  inches.  What  is  its  volume? 
This  is  what  part  of  a  cubic  foot?  ^  of  a  brick  wall  is 
usually  filled  with  mortar.  How  many  bricks  will  be  required 
to  build  a  wall  40  feet  long,  8  feet  high,  and  8  inches  thick? 

40.  Find  the  cost  of  an  8-inch  foundation  for  a  building 
18  feet  X  24  feet,  the  wall  to  be  7  feet  high,  and  brick  $8  a 
thousand  in  the  wall. 

41.  Two  men  start  from  the  same  place  and  travel  in 
opposite  directions,  one  at  the  rate  of  3.85  miles  per  hour, 
and  the  other  at  the  rate  of  4.12i  miles  per  hour.  How  far 
apart  will  they  be  at  the  end  of  13  hours? 


MEASUREMENT  OF  THE   CIRCLE.  143 

42.  A  freight  train  runninG;  at  an  average  rate  of  16| 
miles  au  hour  starts  from  Albany  at  6  a,  m.  At  9  a.  m.  an 
express  train  starts  from  the  same  point  and  runs  iu  the 
same  direction  at  an  average  rate  of  42^  miles  an  hour.  At 
what  time  will  it  overtake  the  freight  train  ?  How  far  from 
Albany  ? 

197.     MEASUREMENT   OF   THE   CIRCLE. 

Measure  accurately  with  a  tape-line  the  circumferences 
and  diameters  of  five  circles,  such  as  the  bottom  of  a  pail  or 
the  head  of  a  barrel.  If  the  figure  measured  is  not  a  true 
circle,  take  half  the  sum  of  the  longest  and  shortest  diame- 
ters as  the  true  diameter.  Divide  each  circumference  by  its 
diameter,  getting  the  quotient  true  to  hundredths,  thus : 
14.5,  ^  4^  ^  14.1S75  ^  4.5  =  3.15+ 

These  quotients  differ  because  of  inaccuracies  in  measure- 
ments.    Find  their  average.     This  is  a  close  approximation 

circumference 

to  the  true  quotient.     This  quotient, -,  is  repre- 

diameter 

seuted  by  the  character  -n-  (called  j*;/). 
Hence  circumference  =  tt  X  diameter. 

PROBLEMS. 

1.  Measure  the  circumferences  of  5  trees  and  calculate  the 
diameter  of  each. 

2.  Measure  the  circumferences  of  5  other  round  objects, 
—  ball,  stove,  apple,  etc.,  —  and  calculate  their  diameters. 

3.  How  long  is  the  tire  on  a  4-foot  wheel? 

4.  How  far  is  a  wheelman  advanced  by  each  revolution  of 
his  28-inch  wheel? 

5.  A  bicycle  is  said  to  be  geared  to  70  inches  when  each 
revolution  of  its  pedals  propels  it  a  distance  equal  to  the 
circumference  of  a  70-inch  wheel.  How  many  revolutions 
of  the  pedals  will  move  the  wheel  one  mile? 


144  NEW  ADVANCED  ARITHMETIC. 

Note.  By  more  accurate  methods  the  value  of  ir  has  been  ivund  to 
200  decimal  places.     Hereafter  call  its  value  3^  or  3.1416—. 

6.  My  bicycle  bas  28-incb  wbeels,  7  sprockets  on  the  rear 
bub,  17  sprockets  at  the  pedal.  How  many  turns  of  the 
pedals  will  carry  me  two  miles  ? 

7.  The  circumference  of  the  earth  at  the  equator  is  24,897 
miles.     What  is  its  equatorial  diameter? 

8.  What  is  the  diameter  of  a  circular  mUe  race- track  ? 

9.  The  wheels  of  a  wagon  are  respectively  42  inches  and 
49  inches.  In  going  what  distance  does  the  fore  wheel  make 
one  more  turn  than  the  hind  wheel? 

10.  How  is  a  circle  drawn  ? 

11.  Draw  a  cu'cle  whose  diameter  is  6  inches ;  one  whose 
radius  is  8  inches.  (The  radius  is  h  of  the  diameter.)  Cal- 
culate the  circumference  of  each.  Cover  the  circumference 
of  each  with  a  string.     Measure  it. 

198.     DEFINITIONS. 

1.  A  Plane  is  a  surface  against  which  a  straight  edge  will 
fit  in  all  directions,  as  the  desk-top,  the  blackboard. 

Point  out  surfaces  in  the  school-room  that  are  not  planes 
nor  made  of  planes.     Such  are  called  curved  surfaces. 

2.  A  circle  is  a  plane  figure  bounded  by  a  line  all  points 
of  which  are  equally  distant  from  a  point  within  called  the 
centre. 

3.  The  bounding  line  is  called  the  circumference.  Any 
straight  line  from  centre  to  circumference  is  called  a  radius. 
A  line  passing  through  the  centre  and  terminating  in  the 
circumference  is  called  the  diameter. 

FORMUL/E. 

D  =  2R.  C  =  7rxDor7rD.         C  =  7rX2Ror27rR. 

NoTK.  When  letters  rei)reseiit  numbers,  the  sign  of  multiplication  may 
be  omitted. 


MEASUREMENT  OF  THE   CIRCLE. 


145 


199.  Cut  from  a  potato 
or  turaip  a  thin  circular 
slice.  Cut  it  iu  ti^-o, 
and  divide  each  semicircle 
into  8  equal  wedges.  Be 
careful  not  to  cut  the 
rind.  Straighten  the  rind 
of  each  semicircle  and  fit 
the  wedges  together.  The 
circle  is  now  a  rhombus. 
Its  base  is  half  the  circum- 

27rR  ^ 

ference,  — - — ,    or    -  K. 


Its  altitude  is  R ;  hence  its  area  is  -  R  X  R  =  tt  R'-. 


NoTK.  An  exponent  is  a  figure  written  above  and  to  the  right  of  a 
figure  or  letter  to  show  liow  many  times  the  number  represented  by  the 
latter  is  to  be  used  as  a  factor ;  thus,  3-  =  3  X  3  ;    4^  =  4  X  4  X  4. 

PROBLEMS. 

1.  If  R  in  the  above  diagram  is  7  inches,  what  is  the 
length  of  AB  ?     What  is  the  area  of  the  circle  ? 

2.  TMiat  is  the  area  of  one  face  of  a  silver  dollar? 

3.  The  face  of  a  watch  is  IJ  inches  in  diameter.  "WTiat 
is  its  area? 

Solution,     it  /?2  =  31  x  (\"T-  =  -V-  X  J"  X  f"  =  2l|  square  inches. 

4.  By  how  many  acres  does  a  square  mile  exceed  a  mile 
circie  ? 

1  mile  =  320  rods-  160  square  rods  =  1  acre. 


146  NEW  ADVANCED  ARITHMETIC. 

5.  "WTiat  is  the  length  of  the  sweat-band  in  a  6|  hat? 

6.  How  high  must  a  3-iuch  tin-cup  be  made  to  hold  one 
pint? 

One  gallon  =  231  cubic  inches. 

7.  Measure  pails,  tin-cups,  coffee-barrels,  and  other  cylin- 
ders and  determine  their  capacity  in  cubic  inches. 

8.  What  is  the  volume  of  a  new  lead  pencil  \"  X  7  "  ? 

9.  Around  a  circular  pond  500  feet  in  diameter  is  a  gravel 
walk  30  feet  wide.     What  is  the  area  of  the  walk? 

Ql-ery.  "What  is  the  area  of  the  entire  circle,  including  walk  and 
pond  ?    the  area  of  the  pond  only  ? 

10.  How  many  square  inches  of  tin  are  needed  to  make  a 
quart  cup  4  inches  in  diameter,  making  no  allowance  for 
seams? 

11.  How  many  barrels  of  3U-  gallons  each  will  a  cylindri- 
cal water-tank  hold  if  14  feet  in  diameter  and  12  ft.,  10  in. 
deep,  inside  measure? 

12.  Mercury  is  shipped  from  the  mines  in  cylindrical  steel 
bottles  holding  100  pounds  each.  If  these  bottles  are  4 
inches  in  diameter,  what  must  be  their  depth? 

Mercury  is  13.6  times  as  heavy  as  water. 

13.  What  is  the  weight  of  a  dry  pine  log  12  feet  long  and 
30  inches  in  diameter?     Specific  gravity  of  dry  pine  =  .48. 

14.  How  many  cubic  inches  in  a  cylindrical  tile  12  inches 
long?     Outside  diameter  8  inches,  inside  diameter  6  inches? 

15.  How  many  square  inches  in  the  entire  surface  of  a 
cylindrical  block  6  inches  in  diameter  and  6  inches  high? 
How  many  cubic  inches  in  its  volume?  Which  is  the  larger, 
surface  or  volume  in  a  4 '^  X  4"  cylinder?  in  an  8"  X  8" 
cylinder  ? 


FEDERAL  MONEY. 


147 


200.    FEDERAL   MONEY. 

1.  Money  is  that  medium  by  which  exchanges  of  property 
are  ordinarUy  effected. 

2.  Federal  Money  is  that  sj'stem  of  mouey  established  by 
the  Congress  of  the  United  States  of  America. 

3.  The  unit  of  value  is  the  dollar. 

4.  AVhat  are  the  denominations?  Give  the  table.  "What 
is  the  scale? 

5.  A  coin  is  a  piece  of  metal  on  which  certain  characters 
are  stamped  by  government  authority,  making  it  legally 
current  as  money. 

6.  The  coins  are  as  follows : 


WEIGHT. 

One  cent    .     . 

Bronze, 

48  grains 

Troy. 

3-ceut  piece    . 

Copper  and  nickel, 

30 

(( 

5-ceut  piece    . 

Copper  and  nickel, 

73.16  " 

(I 

SILVER. 

Dime    . 

Silver  and  copper, 

38  iV  " 

(( 

Quarter-dollar 

>  .                      it 

96^0  " 

(( 

Half-dollar    • 

i;            u 

192.9  " 

li 

Dollar .     .     . 

Ik          a            (( 

412.5  " 

ti 

Gold  and  copper. 


GOLD. 

Quarter-eagle 
Three-dollar  .  "  "  " 
Half-eagle  .  "  "  " 
Eagle  =  SIO  .  "  "  " 
Double-eagle.         "       '•        " 

7.  The  silver  and  gold  coins  are  one  tenth  copper. 

8.  The  gold  dollar  weighs  25.8  grains  Troy.  To  find  the 
weights  of  the  remaining  coins,  multiply  this  number  by  the 
number  of  dollars  expressing  the  value  of  the  coin. 

9.  The  government  also  provides  a  currency  made  of 
paper.  It  consists  of  treasury  notes,  national-bank  notes, 
gold  certiflcates,  and  silver  certificates. 


148  NEW  ADVANCED  ARITHMETIC. 

201.    BILLS    AND    STATEMENTS. 

1.  A  bill  is  an  itemized  statement  of  indebtedness.  "When 
it  includes  items  purcliased  at  different  times,  it  is  usually 
called  "a  statement  of  account." 

When  payment  is  made,  the  statement  is  "  receipted." 

202.    Arrange  the  following  in  bill  forms : 

1.  On  February  10,  1892,  D.  C.  Smith  bought  of  R.  C. 
Rogers  &  Co.,  Rochester,  N.  Y.,  47  rolls  wall-paper,  at  27 
cents;  60  yards  border,  at  3  cents;  13  shades,  at  Si. 72; 
112  feet  moulding,  at  17  cents;  13  sets  curtain  fixtures,  at 
79  cents. 

2.  The  following  is  a  statement  of  W.  P.  Johnson's 
account  with  A.  B.  Cole  &  Co.,  Salem,  Mass.,  made  June 
30,  1891:  1891,  June  15,  2  gallons  molasses,  at  64  cents; 
June  16,  1  sack  flour,  Si. 65;  10  pounds  starch,  at  7  cents; 
9  pounds  turkey,  at  12  cents;  June  21,  5  gallons  oil,  at  15 
cents;  2  loaves  bread,  at  5  cents;  2  dozen  eggs,  at  13  cents, 
June  23,  24  pounds  sugar,  at  5  cents;  June  24,  h  bushel 
potatoes,  40  cents;  June  26,  2  pounds  cheese,  at  20  cents. 
Receipt  this  statement. 

3.  R.  P.  Young,  in  account  with  G.  G.  Johnson,  Normal, 
111.  1891,  December  1,  1  dozen  cakes,  10  cents;  2  loaves 
bread,  at  5  cents ;  1  pound  halibut,  20  cents ;  December  3, 
25  pounds  sugar,  at  4  cents;  1  peck  sweet  potatoes,  25 
cents ;  ^  bushel  apples,  50  cents ;  December  4,  2  barrels 
kindling,  at  25  cents ;  December  7.  3  chimneys,  at  8  cents ; 
1  quart  oysters,  35  cents ;  December  8,  1  pound  tea,  75 
cents;  1  dozen  cakes,  10  cents;  December  10,  2  dozen  eggs, 
at  16  cents;  December  11,  3  lemons,  at  4  cents;  1  sack  salt, 
35  cents ;  December  14,  4  cans  plums,  at  20  cents ;  1  bushel 
apples,  $1.25;  December  17.  1  peck  onions,  28  cents; 
December  20,  3  chickens,  at  25  cents;  5  gallons  oil,  at  15 
cents;  December  24,  12  pounds  turkey,  at  11  cents;  1  quart 
oysters,  35  cents ;  5  bunches  celery,  at  4  cents. 


FEDERAL  MONEY.  149 


203.   ALIQUOT  PARTS. 


6^  cents  is  fj  of  $ 

H      "      iV  "  ^ 

12^         "  i    "    $ 

16|        «         i    "   $ 
25  "         i    "  $ 


33^  ceuts  is  i  of  $1.  66|  cents  is  |  of  $1. 

37^        "        i  "  f  I.  75          "I  "  $1. 

50          '■        ^  "   $1.  83^        "        I  "   $1. 

62^        "        I  "   $1.  87^        "        I  "   $1. 


PROBLEMS. 


1.  What  is  the  cost  of  384  pounds  of  sugar  at  6|  cents 
per  pound  ? 

Analysis.  If  the  sugar  were  $1  a  pound,  384  pounds  would  cost  $384. 
Since  the  sugar  costs  jg  of  a  dollar  a  pound,  384  pounds  will  cost  ^^  of 
$384,  which  equals  $24. 

2.  What  is  the  cost  of  600  articles  at  8^  cents  each? 

3.  Of  688  articles  at  12|  cents  each? 

4.  Of  1,272  articles  at  25  cents  each? 

5.  Of  5,865  articles  at  33|  cents  each? 

6.  Of  575  articles  at  50  cents  each? 

7.  Of  473  articles  at  10  cents  each? 

8.  Of  972  articles  at  20  cents  each? 

Continue  this  exercise  until  great  facility  is  acquired. 

9.  What  is  the  cost  of  568  yards  of  cloth  at  37^  cents  a 
yard? 

Analysis.  If  the  cloth  were  $1  a  yard.  568  yards  would  cost  $568. 
If  the  cloth  were  |  of  $1  a  yard,  568  yards  would  cost  ^  of  $568.  which  is 
$71.  Since  the  cloth  is  f  of  $1  a  yard,  568  yards  will  cost  3  X  $71,  which 
is  $213. 

10.  Of  480  pounds  of  tea  at  62^  cents  per  pound? 

11.  Of  560  articles  at  87^  cents  each? 

12.  Of  1,272  articles  at  66jf  cents  each? 

13.  Of  2,480  articles  at  75  cents  each? 

14.  Of  42,684  articles  at  83^  cents  each? 

IIA 


150 


NEW  ADVAXCED  ARITHMETIC. 


15.  At  12?,  cents  a  dozen,  bow  many  dozens  of  eggs  can 

be  purchased  for  §7o? 

Analysis.  At  $1  a  dozen,  $75  will  buy  75  dozens.  At  ^  of  SI  a 
dozen,  $75  will  buy  8  X  75  dozens,  -which  equals  GOO  dozens. 

16.  At  25  cents  each,  how  many  articles  can  be  bought 
for  $84?  for  8125?  for8M4.oO?  for  8328.75?  for  S875.25? 

17.  At  33 i  cents  each,  how  many  articles  can  be  bought 
for$4U>  for  $i)5?  for6875>  for  8'J75.33i  ?  for  61, 275. 66§  ? 

18.  At  16  5  cents  each,  how  many  articles  can  be  bought 
for  818?  for  $'J6?  for  8324.1G2?  for  $425.33^?  for  $585.50? 
for  8728.66:^  ?  for  82,548.831,  ? 

19.  At  8^  cents  each,  how  many  articles  can  be  pur- 
chased for  25  cents?  for  50  cents?  for  83'  cents?  for  815? 
for  $85?  for  $125.16,2?  for  8250.25?  for  8324.33.1?  for 
$354,415?  for  8681.50?  for  8724.58J-?  for  81,242.66^-?  for 
$1,461.75?  for  8l,5'J5.83i?  for  $2,568,915? 

20.  At  50  cents  each,  how  many  articles  can  be  bought 
for  $50?  for  $125?  for  81,250?  for  $864.50?  for  $965.25? 
for  81,386.75? 

21.  At  37^  cents  a  pound,  how  many  pounds  of  butter 
can  be  bought  for  $15? 

AxALTSis.  At  $1  a  pound,  SI.t  will  l)uy  15  pounds.  At  |-  of  a  dollar 
a  pound,  $15  will  buy  8  X  15  pounds,  which  equals  120  pounds.  At  | 
of  a  dollar  a  pound,  $15  will  buy  ^  of  120  pounds,  which  is  40  pounds. 

22.  At  62i  cents  each,  how  many  articles  will  $125  buy? 
8200?  8645?"  8874.25?  81,025.50?  $2,550? 

23.  At  87 h  cents  each,  how  many  articles  will  $28  buy? 
$42?  $91?  $126;  $357?  $826.42? 

24.  At  66;|  cents  each,  how  many  articles  will  $8  buy? 
$28?  $64?  $186?  $432.50?  $786.48? 

25.  Make  5  problems  in  which  the  cost  of  each  article  is 
75  cents;  83 J  cents;  91§  cents. 


DENOMINATE   NUMBERS.  151 


204.    DENOMINATE  NUMBERS. 

1.  Measuring  is  the  process  of  finding  how  many  times  a 
given  quantity  contains  another  quantity  called  the  unit  of 
measure. 

2.  The  unit  of  measure  is  called  the  Standard  Unit,  and  is 
usually  defined  by  law. 

3.  A  number  composed  of  standard  or  derived  units 
employed  in  the  measuring  of  magnitudes  is  a  Denominate 
Number,  as  3  bushels,  5  pounds. 

4.  A  denominate  number  composed  of  Ijut  one  kind  of 
unit  is  a  Simple  Denominate  Number,  as  3  pecks. 

5.  A  denominate  number  composed  of  more  than  one 
kind  of  unit,  but  which  is  reducible  t  >  a  simjjle  denominate 
number,  is  a  Compound  Denominate  Number,  as  4  bushels, 
3  pecks,  which  is  reducible  either  to  bushels  or  pecks. 

6.  A  scale  is  the  statement  of  the  number  of  units  of  each 
kind  required  to  form  one  of  the  next  higher  kind. 

Note.  We  have  seen  tliat  the  decimal  scale  is  uniform.  Nearly  all  of 
the  scales  in  Compound  Denominate  Numbers  are  not  uniform.  While 
the  scale  in  Federal  Money  is  10,  in  English  Money  it  i.s  4,  12,  20. 

205.    MEASURES    OF    LENGTH. 

1.  A  Line  is  that  which  has  extension  in  only  one  dir'^ction. 

2.  Measures  of  length  are  called  Linear  Measures. 

206.    LINEAR    MEASURE. 
Table. 

12  inches  (in.)  =  1  foot  ^ft.). 
3  ft.  =:  1  yard  (yd.). 

bl  yd.  =  1  rod  ^-d.). 

320  rd.  =  1  mile. 


152  XEW  ADVAXCED  ARITHMETIC. 

Ho"w  many  yards  in  a  mile  ?    How  many  feet  ?    Remember 
these  numbers. 

(Perform  without  analysis.) 

1.  How  many  inches  in  5  feet  ?  7  feet?  8  feet?  15  feet? 
ie\  feet?  18^  feet?  7.8  feet?  5.16  feet? 

2.  How  mauy  feet  in  5  yards?  12  yards?  16^  yards?  18} 
yards?  24  yards?  33. V  yards?  42.6  yards? 

3.  How  many  yards  in  4   rods?  10  rods?  32  rods?  33} 
rods?  37i^  rods?  9.52  rods? 

4.  How  many  rods  in  6  miles?  9  miles?  16.V  nailes?  24^ 
miles  ? 

5.  How  mauy  inches  in  4  feet,  5  inches?  16  feet,  6  inches? 
31  feet,  8  inches? 

6.  How  many  inches  in  5  yd.  1  ft.  6  in.  ?  8  yd.  2  ft.  8  in.  ? 
19  yd.  2  ft.  3.V  in.?  4  rd.  2  yd.  3  in.?  22  rd.  2  ft.  11  in.? 

7.  How  many  feet  in  36  inches?   60  inches?  84  inches? 
100  inches?  115  inches? 

8.  How  many  yards  in  12  feet?  18  feet?  22  feet?  26  feet? 
82  feet? 

9.  How  many  rods  in  11  yards?  33  yards?  44  yards?  38^ 
yards?  49^  yards?  35|  yards? 

10.  How  many  miles  in  640  rods?  960  rods?  1,280  rods? 
1,350  rods?  2,080  rods? 

11.  How  many  feet  and  inches  in  65  inches?  88  inches? 
131  inches?   164  inches?  236  inches? 

12.  How  many  yards,  feet,  and  inches  in  62  inches?  93 
inches?    167  inches?  328  inches?  434  inches? 

13.  How  many  rods,  yards,  and  feet  in  37  feet?  62  feet? 
69  feet?  86  feet? 

14.  How  many  miles,   rods,  and   yards  in   1,797  yards? 
3,569  yards?    19,540  yards? 


■    DENOMINATE  NUMBERS.  153 

207.  Define  reduction.  Define  each  kind.  What  is  the 
form  employed  in  Problem  1,  above?  In  Problem  7?  What 
is  the  general  rule  for  reduction  ascending?  for  reduction 
descending?     (See  Art.  17.) 

Illustrative  Example. 

1.  Reduce  2  mi.  46  rd.  3  yd.  2  ft.  8  in. 

Analysis.  Since  iu  1  mile  there  are  320  rods,  iu  any  number  of  miles 
there  are  320  times  as  many  rods;  hence,  in  2  miles  there  are  320  times 
2  rods,  which  are  640  rods.  640  rods  +  46  rods  =  686  rods.  Since  in  1 
rod  there  are  5i  yards,  in  any  number  of  rods  there  are  5i  times  as  many 
yards;  hence,  in  686  rods  there  are  bh  times  686  yards,  etc. 

2.  Reduce  3  yd.  2  ft.  10  in.  to  inches. 

3.  Reduce  21  rd.  4  yd.  1  ft.  to  feet. 
Reduce  to  feet : 

4  5  rd.  3  yd.  2  ft.  - 

5.  8  rd.  4  yd.  1  ft. 

6.  13  rd.  5  yd.  1  ft. 

7.  38  rd.  2  yd. 

8.  2  mi.  124  rd.  4  yd. 

9.  5  mi.  312  rd.  2  ft. 

10  6  mi.  196  rd.  3  yd.  1  ft. 
Reduce  to  inches : 

11.  3  yd.  2  ft. 

12.  4  yd.  1  fto  8  in. 

13.  5  yd.  2  ft.  10  in. 

14.  4  yd.  2  ft.  5  in. 

15.  2  rd.  4  yd.  7  in. 

16.  1  mile. 

17.  \  mile. 

208     Reduce  to  units  of  lower  denominations  ? 
1.   f  of  a  mUe. 


154  NEW  ADVANCED  ARITHMETIC. 

Analysis.  Since  in  1  mile  tiieie  are  320  rods,  in  f  of  a  mile  there  aw 
f  of  320  ruds  =  ^^^  rods  =  274f  rods,  f  of  a  rod  ~  f  of  V-  yard  =  V  ^^ 
a  yard  =  1  f  yards.  4  of  a  yard  =  f  of  3  feet  =  ^  feet  =  1  j  feet.  ^  of  a 
foot  =  ^  of  12  inches  =  y  iuclies  =  8*  inches;  hence,  etc. 

KUUM. 

f  nil.  =  f  X  -^--'O  rd.  =  LM^-2-^  rd.  =  2742  rd. 

I  rd.  =  f  X  V  y'l-  =  ¥  yd.  =  H  yd. 

^yd.  =  f  X  ;3  ft.  =  Y  ft.  =  If  ft. 
I  ft.    =  f  X  12  ill-  =  Y  =  ^f  i"- 


2 

§  of  a  mile. 

7. 

.625  of  a  rod. 

3. 

^  of  a  rod. 

8. 

,86  of  a  mile. 

4. 

§  of  a  mile. 

9. 

.047  of  a  mile. 

5. 

^\  of  a  yard. 

10. 

.253  of  a  rod. 

6. 

.375  of  a  yanl. 

11. 

.08^  of  4  miles. 

KoTK.     .375  X  3  =  l.l 

125  feet. 

.125 

X  12  =  1.5  inches. 

209.  Reduce  to  units  of  higher  denominations  : 
1.    80526  inches. 

Analysis.  Since  there  are  12  isclies  in  1  foot,  in  80526  inches  there 
are  as  many  feet  as  there  are  I2's  in  80526.  There  are  6710  12's  in  80526, 
with  a  remainder  of  6;  hence,  in  80526  inches  there  are  6710  feet  and  6 
inches.  Since  there  are  3  feet  in  one  yard,  in  6710  feet  there  are  as  many 
yards  as  tliere  are  3's  in  6710.  There  are  2236  3's  in  6710,  with  a  remainder 
of  2;  hence,  in  6710  feet  there  are  2236  yards  and  2  feet.  Since  in  T  rod 
there  are  5^  yards,  in  2236  yards  tliere  are  as  many  rods  as  there  are  times 
5^  yards  in  2236  yards,  or  as  there  are  times  11  lialf-yards  in  4472  half- 
yards.  There  are  406  times  11  in  4472.  with  a  remainder  of  6;  lience.  in 
4472  half-yards  there  are  406  rods,  with  a  remainder  of  6  half-yard^,  or  3 
yards.     Since  there  are  320  rods  in  a  mile,  etc. 

6.  317  feet. 

7.  461  yards. 

8-  2893  inches. 

9-  26459  inches. 

210.  1.  Reduce  f  of  a  foot  to  <^he  fraction  of  a  rod. 


2. 

1253  inches. 

3. 

1367  inches. 

4. 

1598  inches. 

5. 

2291  inches. 

10. 

17891  feet. 

DENOMINATE  NUMBERS.  155 

Analysis.  1  foot  is  ^  of  a  yard.  |  of  a  foot  is  |  of  ^  of  a  yard,  which 
is  2%  of  a  yard.  Since  tliere  are  Y  yards  in  a  rod,  ^  of  a  yard  is  f\  of  a 
rod,  and  1  yard  is  y\  of  a  rod.  ^\  of  a  yard  is  ^^  of  ^j  of  a  rod,  which  is 
yfj  '^f  a  rod.     Short  form.     |  X  J  X  jx- 

2.  6  inches  are  what  part  of  a  rod? 

3.  f  of  a  yard  are  what  part  of  a  mile? 

4.  6  feet  are  what  part  of  a  rod?  of  a  mile? 

5.  ^  of  an  inch  is  what  part  of  a  yard?  of  2  yards?  of  2 J 
yards  ? 

6.  2  feet  3  inches  are  what  part  of  a  yard  ?  of  a  rod  ? 

7.  1  foot  9  inches  are  what  part  of  a  yard?  of  a  rod? 

8.  4  yards  2  feet  are  what  part  of  a  rod  ?  of  a  mile  ? 

9.  I  of  a  foot  are  what  part  of  a  rod  ? 

10.  2\  feet  are  what  part  of  a  rod?  3 J  feet?  4|^  feet? 

11.  1%  of  a  rod  are  what  part  of  a  mile?  2\  rods?  3^ 
rods  ? 

12.  Reduce  to  the  fraction  of  a  rod  .12  of  a  foot;  .015  of 
a  foot?  .08 J  of  a  yard ;   .7  of  an  inch. 

13.  Reduce  .25  of  a  foot  to  the  fraction  of  a  mile;  .375 
of  a  yard;  .875  of  a  rod. 

14.  Reduce  /^  of  a  foot  to  the  fraction  of  2  miles ;  of  3^ 
miles. 

211.    1.    Change  3  yd.  2  ft.  3  in.  to  the  decimal  of  a  rod. 

Analysis.     3  in.  are  J  of  a  foot.     \  =  .25. 

2.25  ft.  =  \  as  many  yds.  =  .75  yd. 
3  75  yd.  =  -^j  as  many  rd.  =  .68+  rd. 

2.  Change  4  yd.  2  ft.  6  in.  to  the  decimal  of  a  rod.  (Re- 
ject all  terms  below  fourth  place.) 

3.  Change  3  rd.  5  yd.  1  ft.  8  in.  to  the  decimal  of  a  mile. 

4.  Change  to  the  decimal  of  a  mile :  4  rods ;  6  rods ;  20 
rods;  10  rods,  3  yards;  50  rods,  5  yards,  2  feet;  80  rods,  2 
yards,  2  feet,  9  inches. 


156  NEW  ADVANCED  ARITHMETIC. 


212.    SURFACE   MEASURE. 

1.  A  surface  is  that  which  has  length  and  breadth  only, 

2.  A  surface  is  measured  by  finding  how  many  surface 
units  it  contains? 

3.  The  surface  unit  is  usually  a  square  whose  side  is  a 
linear  unit. 

4.  A  square  has  the  following  properties : 

(a)  It  is  a  plane. 

(b)  It  is  bounded  by  four  equal  straight  lines. 

(c)  Its  angles  are  all  right  angles. 

5.  How  many  square  inches  in  a  square  foot  ?  square  feet 
in  a  square  yard  ?  square  yards  in  a  square  rod  ? 

6.  Draw  a  square  rod  on  the  scale  of  1  inch  to  the  yard. 

7.  Complete  and  learn  the  following  table. 
square  inches  =  1  square  foot. 

square  feet      =  1  square  yard. 

square  yards   =  1  square  rod. 

160  square  rods     —  1  acre  (A). 

640  acres  =  1  square  mile. 

In  land  surveys  a  square  mile  is  called  a  section. 

213.     PROBLEMS. 

1.  Reduce  2  sq.  rd.  7  sq.  yd.  5  sq.  ft.  86  sq.  in.  to  square 
inches. 

2.  Reduce  3  A.  84  sq.  rd.  10  sq.  yd.  2  sq.  ft.  to  a  simple 
number. 

3.  Reduce  3  sections,  480  A.  125  sq.  rd.  to  square  yards. 

4.  Reduce  12,652  square  inches  to  a  compound  number. 

5.  Reduce  224,725  square  rods  to  square  rods,  acres,  and 
square  miles. 

Note.     The  only  troublesome  divisor  in  square  mejisure  is  30j.     Ob- 
serve the  method  in  the  following  problem. 


DENOMINATE  NUMBERS  157 

€.  Reduce  2,480  square  feet  to  square  rods,  etc 

METHOD. 

2480    square  feet     =    275  sq.  yd.  5  sq.  ft. 
275    square  yards  =1100  fourths  of  a  square  yard. 
30^  square  yards  =  121  fourths  of  a  square  yard.  , 

1100  fourths  -^  121  fourths  =  9,  with  a  remainder  of  11  j 
fourths. 

11  fourths  of  a  sq.  yard  =  2  sq.  yd.  6  sq.  ft.  108  sq.  in. 
Add  first  rem.  5 

3  sq.  yd.  2  sq.  ft.  108  sq.  in. 
Hence,  2480  square  feet  =  9  sq.  rd.  3  sq.  yd.  2  sq.  ft.  108 
sq.  in. 

7.  Reduce  327  square  yards  to  square  rods,  etc. 

8.  Reduce  5,873  square  yards  to  a  compound  number. 

9.  Reduce  -j^f  of  a  square  mile  to  lower  denominations. 

10.  Reduce  f  of  a  square  rod  to  lower  denominations. 

11.  Reduce  .372  of  an  acre  to  lower  denominations. 

12.  Reduce  3.75  square  yards  to  lower  denominations. 

13.  36  square  inches  is  what  part  of  a  square  yard? 

14.  6  A.  64  sq.  rd.  is  what  part  of  a  section? 

15.  Reduce  19  A.  32  sq.  rd.  to  the  decimal  of  a  square 
mile. 

16.  The  base  of  the  Great  Pyramid  of  Gizeh,  which  is  a 
square  764  feet  X  764  feet,  is  what  decimal  fraction  of  a 
square  mile? 

214.    THE   AREAS   OP   RECTANGULAR   SURFACES. 

1.  What  is  the  area  of  a  rectangular  field  40  rods  wide 
and  92  rods  long?     What  is  its  value  at  $63  an  acre? 

2.  What  Is  the  cost  of  plastering  the  walls  and  ceiling  of 
a  room  36  feet  by  48  feet,  and  12  feet  high,  at  27  cents  a 
square  yard,  no  deductions  being  made  for  openings  ? 


158 


NEW  ADVANCED  ARITHMETIC. 


3.  "  Develop  "  the  several  plastered  surfaces  in  Problem  i- 
on  the  scale  of  6  feet  to  1  inch  on  the  blackboard,  or  6  feet 
to  \  incli  on  your  tablet. 

Make  a  similar  diagram  for  each  problem  in  this  set. 


End,  36'. 


.Side,  48'. 


End,  3"^'.      I  Side,  48', 


12' 


Ceiling,  36'  X  48'. 


4.  What  is  the  cost,  at  26  cents  a  square  yard,  of  plaster- 
ing a  cottage  containing  6  rooms,  2  of  which  are  14  feet  by 
15  feet,  2  are  10  feet  by  12  feet,  and  2  are  13  feet  by  15 
feet,  the  ceilings  being  10  feet  high,  and  no  allowance  being 
made  for  openings  ? 

5.  What  will  it  cost  to  paper  the  walls  and  ceilings  of 
this  cottage,  with  paper  at  20  cents  a  roll,  deducting  400- 
square  feet  for  base-board,  allowing  for  14  windows,  each  3 
feet  by  6  feet,  and  for  7  doors,  each  3  feet  by  8i  feet,  the 
paperer's  charge  being  20  cents  a  roll,  and  the  border  costing 
5  cents  a  yard,  no  deductions  being  made  for  border? 

Note.     Wall-paper  is  18  inches  wide.     A  roll  is  8  yards  long. 

6.  Find  cost  of  papering  the  four  walls  and  ceiling  of  your 
school -room  at  5  cents  a  roll,  and  3  cents  per  yard  for  border 
for  walls. 

7.  What  is  the  cost  of  carpeting  a  room  that  is  16  feet  by 
19  feet,  the  carpet  being  a  yard  wide,  and  costing  Si.  12^  a 
yard,  if  there  is  a  loss  of  H  yards  in  matching,  the  carpet 
running  the  longer  dimension  of  the  room? 

Note.  How  many  breadths  are  needed  for  this  room  ?  How  mncli  is 
to  he  turned  nnder  at  one  side  ?  Make  a  plan  of  the  room,  using  a  scale 
of  an  inch  for  a  foot,  and  mark  the  breadths. 


DENOMINATE  NUMBERS.  159 

8.  How  many  yards  of  carpet,  each  strip  being  |  of  a 
yard  wide,  the  loss  in  matching  being  6  inches  in  each  strip 
except  tlie  first,  Avill  be  needed  for  a  room  22  feet  long  and 
18  feet  wide,  the  strips  running  the  long  way?  How  mucli 
if  the  strips  ran  crosswise?  Make  a  plan  of  the  room  for 
each  case.     Which  plan  is  more  economical?     "Why? 

9.  Cost  of  covering  your  school-room  with  cocoa  matting 
at  30  cents  per  square  yard  ? 

10.  Cost  of  carpeting  it  with  ingrain  carpet  36  inches 
wide,  if  the  "  design"  in  the  carpet  is  26  inches  long? 

Note.  How  many  times  is  the  design  repeated  in  one  strip "]  How 
much  must  be  cut  off  or  turned  under  at  the  end  ?  at  the  side  ■? 

11.  If  corn  is  planted  in  hills  4  feet  apart  each  way,  how 
many  hills  in  an  acre  field  9  rods  wide? 

12.  If  corn  is  planted  in  hills  3'-8"  apart  each  way,  and 
yields  3  ears  to  the  hill,  100  ears  to  the  bushel,  what  is  tlie 
yield  in  the  above  field? 

13.  In  a  shower  an  inch  of  rain  fell.  How  many  tons  t® 
the  acre? 

14.  The  water  from  a  roof  37^'  X  56'  is  gathered  mto  a 
cistern  holding  200  barrels  of  31J  gallons.  What  depth  of 
rainfall  on  the  roof  will  fill  the  cistern  ? 

15.  If  the  cistern  is  a  cylinder  10  feet  in  diameter,  an  inch 
of  rain  from  the  above  roof  will  fill  it  to  what  depth  ? 

16  The  pressure  of  the  atmosphere  in  Central  Illinois 
averages  14.4  pounds  to  the  square  inch.  What  is  the  down- 
ward pressure  on  the  lid  of  a  trunk  38  ''  X  21 "? 

215.    SURVEYORS'    MEASURE. 

The  surveyors'  chain,  invented  by  Edmund  Gunter,  about 
1620,  consists  of  100  links.     Its  length  is  4  rods. 

1.  How  many  feet  in  a  chain  ? 

2.  How  many  inches  in  a  link? 


160 


NEW  ADVANCED  ARITHMETIC. 


3.  How  many  chains  in  a  mile? 

4.  Make  a  table  embodying  all  these  facts  and  label  it 
surveyors'  long  measure. 

5.  How  many  square  links  in  a  square  chain?  square 
chains  in  a  square  mile?  square  rods  in  a  square  chain? 
square  chains  in  an  acre? 

6.  Arrange  these  facts  in  a  table  and  label  it  surreyors' 
square  measure. 

7.  How  many  acres  in  a  rectangular  field  22.67  chains  X 
9.26  chains? 

8.  How  many  acres  in  a  rectangular  field  &2  rd.  3  yd.  2  ft. 
3  in.  by  37  rd.  8  in.  ? 

Note   in   the   above    problems   tlie    relative    simplicity   of    surveyors' 


216.    UNITED    STATES    SURVEYS. 

1.  Most  of  the  lands  of  the  United  States  west  of  the  orig- 
inal thirteen  States,  except  the  tract  between  the  Ohio  and 
Tennessee  rivers,  are  surveyed  in  accordance  with  the  fol- 
lowing system : 

2.  In  each  great  survey  district  there  is  run  a  Principal 
Meridian  and  an  east  and  west  line  called  a  Base  Line.     On 

each  side  of  the  Principal  Me- 
ridian at  distances  of  six  miles 
are  north  and  south  lines  called 
Range  Lines,  which  divide  the 
land  into  strips  six  miles  wide 
called  Ranges. 

3.  By  east  and  west  lines  paral- 
lel to  the  Base  Line  the  ranges 
are  divided  into  townships  six 
miles  square.  A  Township  is 
designated  by  giving  its  number 
and  direction  from  the  Base  Line,  the  number  and  position  of 
its  range,  and  the  name  or  number  of  the  Principal  Meridian. 


6 
7 

18 
19 
30 
31 

5 

8 
17 
20 
29 
32 

4 
9 
16 
21 
28 
33 

3 
10 
15 
22 
27 
34 

2 
11 
14 
23 
26 
35 

1 

12 
13 
24 
25 
36 

DENOMINA  TE  NUMBERS. 


161 


Sec.  28. 


\^  Section  28,  etc. 
N.  E.  i,  Section 


Sec- 


3 

1 

4 

2 


Thus  the  writer  is  in  Township  24  North,  Range  2  East  of 
the  3d  Principal  Meridian. 

4.   Townships  are  subdivided  into  36  sections  numbered 
thus: 

The  sections  are  divided  into 
halves  and  quarters  ;  the  quarters 
into  halves  and  quarters,  and  so 
on.     Tracts  are  described  thus  : 

1.  W.  \  Sec.  28,  T.  24  N.  2  E., 
3d  P.  M. 

2.  S.  ^S.E. 

3.  N.  W.  1 
28,  etc. 

4.  W.  i  N.  E.  I  S.  E. 
tion  28,  etc. 

How  many  acres  in  each  of  these  tracts? 
What  fraction  of  a  section  is  each  ? 

217.      PROBLEMS. 
Draw  a  6-inch  square  representing  a  section.     Mark  in  it 
each  of  the  following  tracts  and  tell  how  many  acres  each 
contains. 

1.  The  N.  E.  i  of  the  N.  E.  \. 

2.  The  S.  1  of  the  N.  W.  \. 

3.  The  S.  I  of  the  N.  W.  ^  of  the  S.  W.  \. 

4.  The  E.  i  of  the  N.  \  of  S.  E.  \  of  the  S.  E.  \. 

5.  The  N.  i  of  the  S.  E.  \  of  the  S.  E.  \  of  the  N.  W.  \. 

6.  The  N.  W.  1  of  the  S.  W.  \  of  the  N.  E.  \  of  the  N. 
E.  1. 

The  N.  W.  \  of  the  N.  W.  ^  of  the  N.  ^Y.  \. 
The  N.  W.  1  of  the  N.  W.  \  of  the  N.  E.  \. 
The  N.  ^  of  the  S.  W.  \  of  the  N.  W.  \  of  the 


7. 
8. 
9. 
S.  F 
10 


The  S.  i  of  the  S.  h  of  the  S.  W.  I  of  the  N.  W.  J. 


162  NEW  ADVANCED  ARITHMETIC. 

218.     MEASURES    OF   VOLUME. 

I.  A  Solid  is  a  figure  having  length,  breadth,  and  thick- 
ness. 

2.    A  Rectangular  Solid  is  a  solid 

bounded  by  six  rectangles. 

3.  The  rectangles  are  called  the 
faces  of  the  solid. 

4.  The  intersections  of  the  faces 
are  called  the  edges. 

5.  Name  five  rectangular  solids. 
Bring  a  rectangular  solid  to  the  class. 
Show  the  faces  and  the  edges. 

6.  A  cube  is  a  rectangular  solid  whose  faces  are  squares. 

7.  A  cubic  inch  is  a  cube  whose  edges  are  each  one  inch. 

8.  Solids  are  measured  by  finding  how  many  units  of 
volume  they  contain. 

9.  The  units  of  volume  are  usually  cubes  whose  edges  are 
linear  units. 

10.  Complete  this  table : 

Table  of  Cubic  Measure. 

cubic  inches  =  1  cubic  foot. 

cubic  feet     =  1  cubic  yard. 

128  cubic  feet      =  1  cord  (woOd). 
100  cubic  feet      =  1  cord  (stone). 

II.  Wood  that  is  to  be  used  for  fuel  is  measured  by  the 
cord.  The  sticks  are  usually  cut  4  feet  long,  and  are  then 
called  "  cord-wood."  For  convenience  in  measuring  they 
are  usually  "  corded,"  that  is,  piled  4  feet  high,  the  length 
of  the  stick  making  the  width  of  the  pile.  In  such  a  pile, 
for  every  8  feet  of  length  there  is  a  cord  of  wood.  Prove 
that  it  contains  128  cubic  feet. 


DENOMINATE  NUMBERS. 


163 


A  pile  4  feet  wide,  4  feet  high,  and  1  foot  long  contains  1 
cord  foot.     How  many  cubic  feet  does  it  contain? 

219.      PROBLEMS. 

1.  Reduce  4  cubic  yards  15  cubic  feet  964  cubic  inches  to 
cubic  inches. 

2.  Reduce  8  cords  7  cord  feet  to  cubic  feet. 

3.  Reduce  864,952  cubic  inches  to  a  compound  number. 

4.  Reduce  1,264  cubic  feet  to  cords,  etc. 

5.  Reduce  ^  of  a  cubic  yard  to  cubic  feet  and  cubic  inches. 

6.  Reduce  ^^  of  a  cord  to  integers  of  lower  denominations. 

7.  Reduce  .36  of  a  cord. 

8.  27  cubic  inches  is  what  part  of  a  cubic  yard? 

9.  Change  864  cubic  inches  to  the  decimal  of  a  cord. 


k 


220.     WOOD   MEASURE. 

PROBLEMS. 

1.  How  many  cords  of  cord-wood  in  3  piles  of  wood,  each 
being  4  feet  high,  the  first  being  36  feet  long,  the  second  42 
feet  long,  and  the  third  73  feet  long?  Find  its  cost  at  $4.75 
per  cord. 

Note.  Where  cord-wood  is  piled  4  feet  high,  there  is  cue  cord  for 
every  8  feet  of  length. 


164  NEW  ADVANCED  ARITHMETIC. 

2.  Each  of  the  following  piles  is  4  feet  high.     Find  how 
many  cords  each  contains. 

(a)  83  feet  long.  (d)  47^  feet  long. 

(b)  69  feet  long.  (e)    61  feet  5  inches  long. 

(c)  35^  feet  long.  (/)  93  feet  10  inches  long. 
Find  the  cost  of  each  pile  at  $5.25  a  cord. 

3.  How  much  cord-wood  in  each  of  the  following  piles? 
What  is  the  cost  at  $4.75  a  cord? 

(a)    26  feet  long,  6  feet  high. 


13      3 

g^X0X$4.75       $185.25 
p|  ~        8 


$23.15f. 


Explanation.  The  number  of  cubic  feet  in  this  pile  is 
26  X  6  X  4.     Since  there  are  128  cubic  feet  in  a  cord,  the 

number  of  cords  is Since  cord- wood  is  4  feet 

128 

long,  the  four  may  be  omitted  from  dividend  and  divisor, 

thus  leaving  32  as  a  divisor. 

Note  that  we  then  divide  the  area  of  the  side  of  the  pile  of 
4-foot  wood  by  the  area  of  the  side  of  a  cord. 

(6)    54  feet  long,  7  feet  high. 

(c)  61  feet  long,  7  feet  high. 

(d)  42  feet  8  inches  long,  6  feet  4  inches  high. 

Note.  Call  inches  twelfths  of  a  foot,  and  change  mixed  nnmbers  to 
improper  fractions,  thus : 

128  X  19  X  $4.75 
42  ft.  8  in.  =  42§  ft.  r=  4^.     6  ft.  4  in.  =  6^  =  J^.     3  ^  3  v^  32 

(e)  86  feet  3  inches  long,  7  feet  6  inches  high. 
(/)   124  feet  5  inches  long,  8  feet  6  inches  high. 
(g)    97  feet  6  inches  long,  5  feet  9  inches  high. 
(h)   224  feet  long,  12  feet  high. 


DENOMINATE  NUMBERS,  165 


321      LUMBER    MEASURE. 

1.  The  unit  of  lumber  measure  is  the  board  foot,  one  foot 
long,  one  foot  wide,  one  inch  thick. 

2.  Lumber  that  is  less  than  one  inch  thick  is  counted  as  if 
an  inch  thick.  If  lumber  is  more  than  inch  thick,  the  excess 
is  taken  into  account. 

3.  How  many  cubic  inches  in  a  board  foot?  How  many 
board  feet  in  a  cubic  foot? 

4.  What  is  the  width  of  an  inch  board  that  contains  as 
many  board  feet  as  it  is  feet  long? 

5.  Of  a  two-inch  plank?  of  a  three-inch  stud?   ' 

6.  "What  is  the  end-area  of  each  of  the  above  pieces? 

7.  What  is  the  end-area  of  a  6  "  X  8  "  sill  ?  How  many 
board  feet  in  each  foot  of  its  length?  What  di\isor  have 
you  employed? 

8.  How  many  board  feet  in  a  beam  10 "  x  12  ",  24  feet 
long? 

9.  Show  the  truth  of  the  following : 

thickness  X  •width  X  length 

Number  of  board  feet  = — • 

12 

In   what  units   must    thickness,    width,    and    length    be 

expressed  ? 

Bills  of  lumber  are  regularly  made  out  in  these  units. 

222.       PROBLEMS. 

1.  Wliat  is  the  cost  of  16  sills  6"  X  8"  X  18'  (a  $18  per 
thousand  feet? 

FORM. 

16  X  6  X8  X  18  X  SIS 
12  X  1000 

ANALYSIS.     (1)  Since  a  number  of  dollars  is  required,  I  write  ?18,  the 
cost  of  1000  board  feet.     I  express  the  cost  of  one  board  foot  by  -writing 
1000  as  a  divisor.     I  express  the  cost  of  a  stick  1  foot  long,  1  inch  wide, 
12A 


166  NEW  ADVANCED  ARITHMETIC. 

and  1  inch  thick  by  writing  12  as  a  divisor.     I  multiply  by  18  because  the 

sill  is  18  feet  long;  by  8  because  it  iii  8  inches  wide;  by  6  because  it  is  6 

inches  thick ;  by  1 6  because  there  are  1 6  sills. 

6  X  8  X  18 
(2)  The  number  of  board  feet  in  each  sill  is  expressed  by r^ -. 

1  write  16  as  a  multiplier  because  there  are  16  sills,  1000  as  a  divisor  to 
get  number  of  thousand  feet,  then  multiply  by  18  because  the  number  of 
■dollars  paid  must  be  18  times  the  number  of  thousand  feet. 

2.  What  is  the  cost  of  the  following  bill  of  lumber  at  $21 
•a  thousand  (M)  ?  c 

8  sUls,  8  X  10,  16  feet  long;  48  studs,  2x4,  18  feet 
long;  22  joists,  2  X  10,  16  feet  long;  50  rafters,  2  X  4,  14 
feet  long;  24  joists,  2  X  8,  16  feet  long. 

3.  Find  the  cost  of  the  following  bill  of  lumber,  at  $16.50 
per  M  : 

32  common  boards,  8  inches  wide,  14  feet  long ; 
65  fence  boards,  6  inches  wide,  16  feet  long; 
16  corner  posts,  4x4,  18  feet  long ; 
7  sills,  6  X  8,  14  feet  long; 
46  rafters,  2  X  6,  16  feet  long; 
36  joists,  2  X  8,  18  feet  long. 

4.  Find  the  cost  of  the  following  bill  of  lumber  : 

86  pieces  maple  flooring        1x3    x  16  @  $40  per  M. 
86      "      ash  "  1  X  3^  X  16  @  $36  per  M. 

72      >'      clear  pine  boards  1  X  10  x  16  @  $32  per  M. 
24  rafters  2x6    X  16  @  $18  per  M. 

36  joists  2  X  10  X  22  @  $21  per  M. 

5.  What  is  the  cost  of  46  planks,  2  inches  thick,  10  inches 
wide,  and  18  feet  long,  at  $22  per  M.? 

6.  What  is  the  cost  of  65  2^-inch  oak  planks,  12  inches 
wide  and  16  feet  long,  at  $36  per  M.  ? 

7.  How  many  posts  set  8  feet  apart  are  required  for  wire 
fencing  around  a  field  800  feet  square  and  for  two  cross 
fences  dividing  the  field  into  four  equal  squares  ?  Show  that 
the  cost  of  these  posts  at  8\  cents  each  is  $49.75. 


DENOMINATE  NUMBERS.  167 

8.  Find  the  cost  of  the  lumber  and  posts  to  fence  a  field 
AO  rods  by  60  rods  with  a  5-board  fence ;  the  posts  costing 
22  cents  each,  and  placed  8  feet  apart;  the  boards  being  16 
feet  long  and  6  inches  wide,  and  costing  818.50  per  M. 

Make  a  plan  of  the  field.  Cut  boards  so  as  to  have  as 
little  waste  as  possible. 

9.  With  lumber  and  posts  as  in  the  preceding  problem, 
find  the  cost  of  material,  except  the  nails,  to  fence  the  N.  ^ 
of  N.  W.  \  of  S.  E.  J  of  a  section. 

10.  Find  the  cost  of  lumber  and  posts  to  put  a  4-board 
fence  around  a  section  of  land  and  to  make  division  fences 
separating  it  into  quarter-sections,  boards,  and  posts  as  in 
Problem  8. 

223.     MEASURES  OF  CAPACITY. 

1.  The  units  of  Liqmd  Measure  are  the  gallon,  the  quart, 
the  pint,  and  the  gill.  The  primary  unit  is  the  gallon,  and 
contains  231  cubic  inches.  The  other  u-nits  are  divisions  of 
the  gallon. 

TABLE. 
4  gUls  (gi.)  =  1  pint  (pt). 
2  pt.  =1  quart  (qt.) 

4  qt.  =1  gallon  (gal.). 

Note.  The  pint  is  divided  iDto  16  eqnal  parts,  each  of  which  is  called 
an  ounce.  This  mea,«ure  is  used  by  apothecaries.  The  teacher  should 
show  a  1 -ounce,  a  2-ounce,  and  a  4-onnce  bottle. 

In  computing  the  capacity  of  tanks  and  cisterns,  the  barrel  of  31  §  g^- 
lons  is  tlie  unit. 

2  The  units  of  Dry  Measure  are  the  bushel,  the  peck,  the 
quart,  and  the  pint. 

The  primary  unit  is  the  bushel,  which  contains  2,150|  cubic 
inches.     The  other  units  are  divisions  of  the  bushel. 

Note.  The  standard  bushel  is  18i  inches  in  diameter  and  8  inchefl 
deep.     It  contains  2,1501  cubic  inches,  or  nearly  1^  cubic  feet. 


168  NEW  ADVANCED  ARITHMETIC. 

TABLE. 
2  pints     =  1  quart. 
8  quarts  =  1  peck  (pk.). 
4  pk.       =  1  bushel  (bu.) 
3.   How  many  cubic  inches  are  there  in  the  liquid  quart? 
in  the  dry  quart?     About  how  many  gallons  are  there  in  a 
cubic  foot  of  water? 

PROBLEMS. 

1.  Reduce  3  gal.  3  qt.  1  pt.  2  gi.  to  gills. 

2.  Reduce  17  bu.  3  pk.  5  qt.  1  pt.  to  pints. 

3.  Reduce  587  gills  to  a  compound  number. 

4.  Reduce  1,267  pints  to  a  compound  number. 

5.  Reduce  to  integers  of  lower  denominations  ^-|   of   a 
gallon. 

6.  Reduce  ||  of  a  bushel  to  integers  of  lower  denomina- 
tions. 

7.  ^  of  a  pint  is  what  part  of  a  bushel? 

8.  §  of  a  gill  is  what  part  of  a  gallon? 

9.  Reduce  .625  of  a  bushel  to  integers  of  lower  denomina- 
tions. 

10.  Reduce  1  gill  to  the  decimal  of  a  gallon. 

224.     WEIGHT. 

1.  Weight  is  the  measure  of  the  downward  pressure  of 
bodies  at  or  near  the  surface  of  the  earth. 

2.  There  are  four  systems  of  weight.     These  are  Avoir- 
dupois, Troy,  Apothecaries',  and  Metric. 

3.  The  standard  from  which  the  units  of  the  first  three 
are  derived  is  the  Troy  [)ound  of  the  mint. 

Note  1.     For  the  metric  system,  see  Appendix. 


I 


DENOMINATE  NUMBERS.  169 

4.  The  Troy  pound  is  divided  into  5,760  equal  parts  called 
grains.  7,000  grains  equal  a  pound  avoirdupois,  and  5,760 
the  pound  apothecaries'. 

5.  Troy  weight  is  used  in  measuring  gold,  silver,  precious 
stones,  jewels,  etc. 

6.  Apothecaries'  weight  is  used  in  mixing  medicines. 

7.  Avoirdupois  weight  is  used  in  measuring  ordinary  arti- 
cles of  merchandise. 

8.    Avoirdupois  Weight. 
TABLE. 
16  oz.     =■  1  lb. 
2©00  lb.     =  1  ton  (T.). 

Note  1 .  The  long  ton  is  used  in  the  United  States  custom-houses  and 
in  the  Eastern  States  in  weighing  coal  and  iron. 

28  lb.      —  1  quarter  (qr.). 
4  qr.     =  1  hundred  weight  (cwt.). 
20  cwt.  =  1  tju. 

Note  2.     The  relation  of  Avoirdupois  weight  to  Troy  weight  may  be 
seen  by  comparing  the  following  table  with  the  Troy  table : 
y\  of  7000  grains  =  437j  grains  =  1  oz.  av. 
jig  of  437^  grains  =  2'\\  grains  =  1  dram  av. 
Note  3.     62^  lb.  Avoirdupois  =  1000  oz.  =  the  weight  of  a  cubic  foot 
of  distilled  water. 

9.    Troy  Weight. 
TABLE. 
24  gr.     =  1  pennyweight  (pwt.). 
20  pwt.  =^  1  oz. 
12  oz.     =  1  lb. 

10.     Apothecaries'  Weight. 

TABLE. 
20  gr.  =  1  scruple  (9). " 

3  scruples  =  1  dram  (3). 

8  drams      =  1  ounce  (§). 
12  ounces    =  1  pound  (ib). 


170  NEW  ADVANCED  ARITHMETIC. 

Note.  The  Troy  pound  is  little  used.  Gold  and  silver  buUion  are  sold 
by  the  ounce;  gold  ornaments  by  the  pennyweight;  jewels  by  the  caraZ 
(3.2  grain.s). 

The  word  carat  is  also  used  in  the  sense  of  twenti/-fourths  in  stating  the 
purity  of  gold.     Grold  14  carats  fine  is  ^|  gold,  ^|  alloy. 

The  metric  system  is  rapidly  displacing  Apothecaries'  weight  in  phar- 
macy. 

11.    Comparison  of  Weights. 

1  lb.  Troy  =  f  J§g  =  i^i  of  1  lb.  Avoirdupois. 
1  oz.  Troy  =  ^§^^  =  ff  §  of  1  oz.  " 

The  ounce  and  pound  Apothecaries'  equal  the  ounce  and 
pound  Troy,  respectively. 

225.     PROBLEMS. 

1.  How  many  ounces  in  8  lb.  7  oz.  av.  ?  in  37  lb.  13  oz.? 
in  548  lb.  15  oz.  ?  in  J  of  1  lb.  ?  in  f  of  1  T.  ?  in  .325  of  a 
long  ton? 

2.  How  many  grains  in  2  lb.  5  oz.  11  pwt.  16  gr.  Troy? 
in  3  lb.  8  pwt.  ?  in  H  of  1  lb.  ?  in  .0875  of  1  lb.  ? 

3.  How  many  grains  in  2  lb.  3  oz.  I  sc.  17  gr.  Apotheca- 
ries'? in  -1^^  of  1  lb.?  in  .28  of  1  oz.? 

4.  Change  the  following  simple  numbers  to  compound 
numbers:  568  oz.  av. ;  2825  gr.  Troy;  6827  gr.  Apotheca- 
ries' ;   174  sc. ;  869  pwt. 

5.  What  is  the  cost  of  2  tons  of  sugar  at  3J  cents  a 
pound?  of  a  3-ounce  silver  watch-case  at  8.6  cents  a  penny- 
weight? of  2  oz.  5  gr.  of  quinine  at  J  of  a  cent  a  grain? 

6.  2  lb.  12  oz.  is  what  part  of  1  T.  ? 

7.  Reduce  6  grains  Troy  to  the  decimal  of  a  pound. 

8.  2  sc.  12  gr.  is  what  part  of  2  lb.  5  dr.  ? 

9.  Reduce  4.28  lb.  Troy  to  numbers  of  lower  denomina- 
tions. 

10.  /j  of  1  lb.  Apothecaries'  =? 
U.    12  oz.  is  what  part  of  1  T.  ? 


DENOMINATE  NUMBERS.  XII 

12.  Change  6  gr.  to  the  decimal  of  a  Troy  pound. 

13.  Change  15  lb.  Avoirdupois  to  Troy  weight. 

14.  Change  24  lb.  Troy  to  Avoirdupois  weight. 

15.  Change  22  lb.  13  oz.  Avoirdupois  to  Troy  weight. 

16.  Change  18  lb.  7  oz.  Troy  to  Avoirdupois  weight. 

17.  What  are  you  worth  if  you  are  worth  your  weight  in 
gold  coin  ?  in  silver  dollars  ? 

18.  "A  pint  is  a  pound  the  world  around."  Is  this 
statement  exactly  true  for  water? 

19.  The  Orloflf  diamond  (194|  carats)  weighs  how  many 
ounces  Avoirdupois? 

20.  What  is  the  value  of  the  gold  in  an  18-carat  watch- 
case  weighing  3  ounces  Troy?     1  oz.  gold  =  $20.67. 

Note.  The  grains  are  usually  measured  by  weight.  The  weight  of 
the  bushel  is  determiaed  by  law.  These  laws  are  not  absolutely  uniform 
in  the  several  States,  except  in  the  case  of  wheat.  The  following  is  the 
number  of  Avoirdupois  pounds  for  a  bushel  in  the  great  majority  of  cases: 

Indian  corn,  56 ;  oats,  32 ;  rye,  56 ;  wheat,  60. 

21.  How  many  bushels  in  8,640  lbs.  of  wheat?  In  7,924 
lbs.  of  corn?     In  5,872  lbs.  of  oats?     In  10,240  lbs.  of  rye? 

226.     Apothecaries'  Fluid  Measure. 

60  minims  (m.)    =  1  fluid  drachm  (fl.  3). 

8  fluid  drachms  =  1  fluid  ounce  (fl.  §). 
16  fluid  ounces     =  1  pint  (O.). 

8  pints  =  1  gallon  (cong.). 

Compare  this  table  with  Apothecaries'  Weight. 

227.      PROBLEMS. 

1.  How  many  20-minim  doses  of  laudanum  in  a  fluid 
ounce  ? 

2.  How  many  3-ounce  bottles  of  perfumery  may  be  filled 
from  one  gallon  ? 


172  NEW  ADVANCED  ARITHMETIC. 

3.  A  stationer  bought  5  gallons  of  ink  for  $3.00.  He 
bought  -i-ounce  bottles  ai  10  cents  per  dozen,  filled  them 
with  ink,  and  sold  them  at  5  cents  each.  What  was  his 
gain? 

228.  ENGLISH   MONEY. 

4  farthings  (qr.)  =  1  penny  (d.). 
12  pence  =  1  shilling  (s.). 

20  shillings  =  1  pound  (£). 

£=  Zi'ftra  =  pound,  d.  —  denai'ius^  Latin  for  "penny." 
qr.  =  quadrans  =  fourth. 

Note.  The  Troy  pound  of  silver  was  originally  coined  int.o  240  silver 
pennies  of  24  grains  (1  pennyweight)  eacli.  A  cross  was  stamped  so  deep 
that  the  penny  was  readily  broken  into  fourths  (farthings). 

The  present  value  of  the  pound  sterling  is  $4.8665.  The  gold  coin  of 
this  value  is  called  the  sovereign.  The  shilling  is  coined  of  silver ;  the 
penny  and  half-penny  of  copper.  The  guinea  (21  s.)  and  crown  (5  s.)  are 
no  longer  coined.     English  gold  coins  are  22  carats  fine. 

PROBLEMS. 

1.  Reduce  £7  15  s.  7  d.  1  qr.  to  farthings. 

2.  Reduce  £28  9  d.  to  pence. 

3.  Reduce  586  d.  to  a  compound  number. 

4.  tS^  of  £1  =  what? 

5.  £  7.048  =  what? 

6.  7  d.  2  qr.  is  what  part  of  £1  ? 

7.  Change  15  s.  9d.  to  the  decimal  of  a  pound. 

8.  By  how  much  does  the  quarter  dollar  exceed  the  shilling 
in  value  ? 

229.  FRENCH   MONEY. 

The  franc  (worth  19.3  cents  in  U.  S.  money)  is  the  unit. 
The  scale  is  decimal. 

TABLE. 

lO  millimes  (m.)  =  1  centime  (c.) 
10  centimes  =  1  decime. 

10  decimes  =  1  franc. 


DENOMINATE   NUMBERS.  173 

230.     GERMAN   MONEY. 
The  unit  is  the  inark,  or  reichsmark^  worth  23.8  cents  in 
U.  S.  money. 

TABLE. 
100  pfeunige  (pf.)  =  1  mark  (RM.). 
The  5-franc  piece  has  been  proposed  as  an  international 
coin  nearly  equal  to  the  dollar,  to  4  shillings,  and  to  4  marks. 

1.  What  part  of  the  dollar  must  be  takeu  out  to  make  it 
conform  to  the  5-franc  piece  ? 

2.  What  part  of  4  shillings  ? 

3.  The  weight  of  the  4  marks  must  be  increased  by  what 
part  of  itself? 

231.     CIRCULAR   MEASURE. 

1.  An  arc  is  any  portion  of  a  circumference. 

2.  The  unit  of  arc  measurement  is  the  degree  (°).  A 
degree  is  ^^^  of  a  circumference.  What  part  of  the  circum- 
ference is  an  arc  of  90=  ?  60= ?  45°?  270=?  30?  15°? 
22i=? 

Note.  The  arc  and  the  degree  that  measures  it  must  be  portions  of 
the  same  circumference. 

3.  An  arc  of  90°  is  called  a  quadrant ;  an  arc  of  60=  a 
sextant ;   an  arc  of  30=  a  sign. 

4.  The  degree  is  divided  into  60  equal  parts  called  min- 
utes  (') ;  the  minute  into  60  equal  parts  called  seconds  ("). 

5.  Make  a  table  setting  forth  these  facts  and  label  it 
Table  of  Circular  Measure. 

6.  An  angle  ifi  the  difference  in  direction  between  two 
lines  proceeding  from  the  same  point,  called  the  vertex. 

7.  If  the  four  angles  formed  by  two  intersecting  straight 
lines  are  equal,  each  angle  is  called  a  right  angle.  r 

8.  An  angle  larger  than  a  right  angle  is  an  obtuse  angle, 
an  angle  smaller  than  a  right  angle  is  an  acute  angle. 


174  NEW  ADVANCED  ARITHMETIC, 

9.  The  unit  of  angular  measurement  is  called  a  degree. 
This  degree  is  one  ninetieth  of  a  right  angle. 

Note.  If,  with  the  vertex  of  a  right  angle  as  a  center,  and  any  radius 
whatever,  a  circumference  be  described  intersecting  the  sides  of  the  angle, 
the  intercepted  arc  is  a  quadrant  (90°).  Hence  a  right  angle  is  called  an 
angle  of  90°.  It  is  evident  that  any  acute  angle  is  as  many  ninetieths  of 
a  right  angle  as  its  intercepted  arc  is  of  a  quadrant.  For  this  reason  the 
term  "  degree  "  is  applied  both  to  the  ninetieth  of  the  ri<5ht  '(ngle  and  the 
ninetieth  of  the  quadrant. 

PROBLEMS. 

1.  Reduce  68°  45'  36"  to  seconds. 

2.  Reduce  46824"  to  a  compound  number. 

«.  Add  42°  17'  26",  51^  48'  51",  7°  56'  48",  12°  46'  28'\ 

4.  What  is  the  difference  between  29°  12'  42"  and  16°  46' 
25"  ? 

5.  Multiply  12°  7'  18"  by  16. 

6.  Divide  49°  17'  24"  by  12. 

7.  5"  is  what  part  of  a  degree  ? 

8.  T^^of  1°=? 

232.     LONGITUDE   AND   TIME. 

1.  Find  in  your  atlas  a  map  of  the  world  in  hemispheres. 
What  are  the  lines  called  that  extend  from  the  top  of  the 
map  to  the  bottom?  What  are  they  for?  What  are  the 
cross  lines  called?     What  are  they  for? 

AYhat  is  a  prime  meridian  ?  AVhat  two  prime  meridians 
are  used  in  your  geography? 

2.  Find  a  map  of  the  United  States.  Find  the  longitude 
of  the  following  places  with  reference  to  Greenwich  and  to 
Washington  City : 

(1)  Cape  Cod,  Mass.  (4)    Denver,  Col. 

(2)  Erie,  Pa.  (5)    Leavenworth,  Kan, 

(3)  Washington,  D.  C.  (6)    Memphis,  Tenn. 


DENOMINATE  NUMBERS.  175 

3.  The  longitude  of  a  place  is  its  distance  in  degrees, 
minutes,  and  seconds,  east  or  west  of  a  prime  meridian.  It 
is  measured  on  the  arc  of  a  parallel,  or  of  the  equator. 

4.  The  difference  of  longitude  of  two  places  on  opposite 
sides  of  the  prime  meridian  is  the  sum  of  their  respective 
longitudes. 

5.  If  A  is  10  miles  east  of  B,  and  C  is  12  miles  west  of 
B,  then  A  is  how  many  miles  east  of  C?  How  found?  If 
A  is  5  miles  west  of  B,  and  C  is  21  miles  west  of  B,  then  C 
is  how  many  miles  west  of  A  ?  How  found  ?  Apply  these 
illustrations  in  the  following  problems? 

The  Greenwich  meridian  is   referred  to  in  the  following : 

233.      TABLE. 

Paris 2°  20'  22"  E. 

London 0°    5'  38"  W. 

New  York 71°    0'    3"  W. 

Boston      , 71°    3'  30"  W. 

Chicago     o 87°  35'    0"  W. 

New  Orleans       .     .     .     .  •  .  90°    3'  28"  W. 

San  Francisco    .     .     .     ,     .  122°  26'  15"  W. 

Berlin 13°  23'  43"  E. 

St.  Petersburg   .     .     .     .     .  30°  16'    0"  E. 

Pekin 116°  26'    0"  E. 

Calcutta   .     , 88°  19'    2"  E. 

Pittsburg .......  80°    2'    0"  W. 

St.  Louis 90°  12'  11"  W. 

Cincinnati 84°  26'    0"  W. 

Rome 12°  27'  14"  E. 

Honolulu 157°  52'    0"  W. 

Sydney 151°  11'    0"  E. 

PROBLEMS. 
1.   What  is  the  difference  between  the  longitude  of  Paris 
and  that  of  New  York? 


176  NEW  ADVANCED  ARITHMETIC. 

2.  Of  Berlin  and  of  London? 

3.  Of  Chicago  and  Calcutta? 

4.  Of  Sydney  and  Honolulu? 

5.  Of  St.  Petersburg  and  of  St.  Louis? 

6.  Of  Rome  and  of  Cincinnati? 

How  find  the  difference  of  longitude  of  two  places  on 
opposite  sides  of  the  prime  meridian  ?     Why  ? 

7.  Of  Boston  and  of  Chicago? 

8.  Of  London  and  of  New  Orleans? 

9.  Of  Pekin  and  of  Calcutta? 

10.  Of  Pitts'ourg  and  of  San  Francisco? 

How  find  the  difference  of  longitude  of  two  places  on  the 
same  side  of  the  prime  meridian  ?     Why  ? 

Longitude  to  Time. 

234.  It  is  noon  at  any  place  when  the  sun  is  on  its 
meridian.  Before  it  is  noon  again  at  that  place  the  earth 
must  make  about  one  revolution  on  its  axis.  The  time 
occupied  in  making  this  revolution  is  divided  into  24  hours, 
and  is  called  one  day.  During  this  time  the  entire  circum- 
ference of  each  parallel  has  passed  under  the  sun.  Since 
each  circumference  contains  360°,  ^  of  360°,  or  15°,  passes 
under  the  sun  each  hour,  j.}j^  of  15°,  or  15',  each  minute,  and 
b\j  of  15',  or  15",  each  second. 

When  it  is  noon  at  any  place,  what  time  is  it  15°  east  of 
that  place?  30°  E.?  45°  E.?  90°  E.?  180°  E.?  15°  we^t? 
30°  W.?  45°  AV.?  90°  W.?  120°  W.?  180°  W.?  15' E.?  15' 
W.  ?  30°  30'  E.  ?  45°  45'  W.  ?  60°  30'  15"  E.  ?  00°  45'  30"  W.  ? 

235.  Prove  that  the  following  statements  are  true: 

A  difference  of  15°  in  the  longitude  of  two  places  makes 
a  difference  of  one  hour  in  their  time. 

A  difference  of  15'  in  the  longitudes  of  two  places  makes 
a  difference  of  one  minute  in  their  time. 

A  difference  of  15"  in  the  longitude  of  two  places  makes 
a  difference  of  one  second  in  tlieir  time. 


DENOMINATE  NUMBERS.  Yll 

PROBLEMS. 

1.  "Wliat  is  the  difference  in  time  of  two  places  whose 

difference  of  longitude  is  36°  42'  30"  ? 

Analysis.  Since  a  difference  of  15°  in  the  longitude  of  two  places 
makes  a  difference  of  1  hour  in  their  time,  a  difference  of  36°  of  longi- 
tude makes  a  difference  of  2  hours  in  their  time,  with  a  remainder  of 
6°.  6°  =  360'.  360'  +  42'  =  402.'  Since  a  difference  of  15'  in  the  longi- 
tude of  two  places  makes  a  difference  of  1  minute  in  their  time,  a  dif- 
ference of  402'  of  longitude  makes  a  difference  of  26  minutes  of  time,  with 
a  remainder  of  12'  of  longitude.  12'  =  720".  720"  +  30"  =  750".  Since 
a  difference  of  15"  of  longitude  makes  a  difference  of  1  second  of  time,  a 
difference  of  750"  of  longitude  makes  a  difference  of  50  seconds  of  time. 
Therefore,  a  difference  of  36°  42'  30"  of  longitude  makes  a  difference  of 
2  hr.  26  min.  50  sec.  in  time. 

Find  the  difference  of  time  when  the  difference  of  longi- 
tude is : 

2.  94°  17' 45".         5.      64°    0' 50".         8.     82°  31' 30". 

3.  112°  48' 15".         6.  48' 45".         9.      59°  59' 48". 

4.  6°  56' 46".         7.    150°  12' 42".       10.    128°  19' 18". 

Time  to  Longitude. 

236.  If  the  time  at  A  is  an  hour  later  than  at  B,  what  is 
their  difference  of  longitude?  if  2  hours  later?  if  5  hours 
later?  if  1  minute  later?  5  minutes?  10  minutes?  1  second? 
5  seconds?  20  seconds?  A  is  east  or  west  of  B?  How  do 
you  kno^v? 

Change  the  word  later  to  earlier,  and  ask  the  same 
questions. 

237.  Prove  the  truth  of  the  following  statements : 

A  difference  of  an  hour  in  the  times  of  two  places  shows 
a  difference  of  15°  in  their  longitudes. 

A  difference  of  a  minute  in  the  times  of  two  places  shows 
a  difference  of  15'  in  their  longitudes. 

A  difference  of  a  second  in  the  times  of  two  places  shows  a 
difference  of  15"  in  their  longitudes. 

12 


178  NEW  ADVANCED  ARITHMETIC. 

PROBLEMS. 

1.  The  time  in  one  town  is  2  hr.  35  min.  22  sec.  earlier  than 
in  another.    Which  is  farther  east  ?    How  many  degrees,  etc.  ? 

Analysis.  Since  the  time  is  earlier  in  the  first  town  than  in  the  sec 
ond,  the  sun  will  not  reach  its  meridian  until  it  has  passed  the  meridian  ot 
the  other;  it  is,  consequently,  farther  west.  Since  a  difference  of  1  sec- 
ond in  the  time  of  two  places  shows  a  difference  of  15"  in  their  longitude, 
a  difference  of  22  seconds  in  their  times  shows  a  difference  of  330"  of 
longitude,  which  equals  5'  30".  Since  a  difference  of  1  minute  in  the 
times  of  two  places  shows  a  difference  of  15'  in  their  longitude,  a  differ- 
ence of  35  minutes  in  their  times  shows  a  difference  of  525  in  their  longi- 
tude. 525'  +  5'  =  530'  =  8°  50'.  Since  a  difference  of  1  hour  in  the 
times  of  two  places  shows  a  difference  of  15°  in  their  longitude,  a  differ- 
ence of  2  hours  in  their  times  sliows  a  difference  of  30°  in  their  longi- 
tude.    30°  +  8°  =  38°.     Their  difference  in  longitude  is  38°  50'  30". 

Why  begin  with  the  lowest  denomination  ? 

In  the  following  problems  A  and  B  represent  places  the 
difference  of  whose  times  is  given.  Find  their  difference  of 
longitude,  and  tell  which  is  farther  east. 

2.  4  hr.  25  min.  15  sec.     A's  later. 

3.  8  hr.  6  min.  20  sec.     B's  later. 

4.  6  hr.  40  min.  18  sec.     A's  earlier. 

5.  9  hr.  52  min.  3  sec.     B's  earlier. 

6.  1  hr.  59  min.  59  sec.     A's  later. 

7.  12  hr.     B's  earlier.        n 

8.  2  min.  3t  sec.     L"?  later. 

9.  11  hr.  24  sec.     A's  earlier. 

10.  10  hr.  31  min.  29  sec.     A's  later. 

Make  a  rule  for  each  of  the  two  general  processes. 

11.  When  it  is  noon  at  Paris,  what  is  the  time  at  St. 
Petersburg?  at  San  Francisco? 

12.  When  it  is  6  A.  m.  at  London,  what  is  the  time  at  New 
York?  at  Cincinnati?  at  Rome?  at  Honolulu?  at  Sydney? 


DENOMINATE  NUMBERS.  179 

13.  "When  it  is  35  minutes  past  3  p.  m.  at  Berlin  it  is  34 
min.  41 1'^  sec.  past  8  p.  m.  at  a  second  city.  Find  the  name 
of  the  city  in  the  table. 

14.  When  it  is  4  p.  m.  at  Chicago  it  is  40  min.  35  sec.  past 
1  Po  M.  at  a  second  city.     Find  its  name  in  the  table. 

15.  A  ship's  chronometer  indicates  that  the  time  at  Green- 
wich is  25  minutes  past  3  p.  m.  By  observations  the  captain 
ascertains  that  it  is  noon  where  the  ship  is.  What  is  the 
longitude  of  the  ship? 

Note.  The  teacher  may  form  inauy  problems  from  the  table  of 
longitudes. 

238.    A  Shorter  Method. 

Since  a  difference  of  15°  in  the  longitude  of  two  places 
makes  a  difference  of  one  hour  in  their  times,  a  difference 
of  1°  in  their  longitude  makes  a  difference  of  4  minutes,  and 
a  difference  of  1'  a  difference  of  4  seconds  in  their  times. 
The  two  sets  of  facts  may  be  combined  for  rapid  oral  work. 

Illustration.     Difference  of  longitude  49°  36'  21". 

Analysis.  A  difference  of  45°  makes  a  difference  of  3  hours.  A  dif- 
ference of  4°  makes  a  difference  of  16  minutes.  A  difference  of  30'  makes 
a  difference  of  2  minutes.  A  difference  of  6'  makes  a  difference  of  24 
seconds.  A  difference  of  21"  makes  a  difference  of  If  seconds.  Com- 
bining results,  the  difference  of  time  is  found  to  be  3  hr.  18  min.  25|  sec. 


Solve  Problems  2-10  (page 
177)  by  this  method. 


FORM. 

49° 

36' 

21" 

3 

16 

24 

2 

If 

3  hr.  18  min.  251  sec. 


239.  Apply  the  same  facts  to  the  method  of  finding  differ- 
ence of  longitude  when  difference  of  time  is  given. 

Illustration.  Difference  of  time  5  hr.  39  min.  50  sec.  A 
difference  of  4  seconds  shows  a  difference  of  1',  and  a  differ- 
ence of  4  minutes  a  difference  of  1°. 


180  NEW  ADVANCED  ARITHMETIC. 

Analysis.  A  difference  of  5  hours  shows  a  difference  of  75°.  A  dif- 
ference of  36  minutes  (9  fours  of  minutes)  shows  a  difference  of  9°,  and  of 
3  minutes  a  difference  of  45'.  A  difference  of  48  seconds  shows  a  differ- 
ence of  12'  and  of  2  seconds  a  difference  of  30".  Combining  results,  the 
difference  of  longitude  is  84°  57'  30". 


Solve  Problems  2-15  in 
the  last  set  by  this  method. 


FORM. 

5hr. 

39 

min. 

50  sec. 

75° 

9° 

45' 

12' 

30" 

84°       57'  30' 


240.     THE    INTERNATIONAL    DATE    LINE. 

Travellers  across  the  Pacific  Ocean  westward  set  their  time  forward  a 
dav  on  crossing  the  180th  'r.eridian.  Islands  in  the  equatorial  portion  of 
the  Pacific  were  colonized  by  Europeans  coming  from  the  east  with  the 
trade  winds,  and  have  the  same  reckoning  as  the  American  continent. 
Australia,  New  Zealand,  and  the  neighboring  islands  originally  colonized 
by  the  Dutch  have  the  time  of  Asia,  one  day  in  advance.  On  many  charts 
is  shown  the  International  Date  Line  separating  these  lands.  It  passes 
through  Behring's  Strait,  thence  southwest  east  of  Japan,  but  west  of  the 
Philippines,  thence  east,  southeast,  and  south  to  the  east  of  New  Zealand- 
Prior  to  1867  this  line  passed  east  of  Alaska. 

241.     TIME    MEASURES. 
60  seconds  (sec.)  make  1  minute  (min.). 


60  minutes 
24  hours 
7  days 

365  days 

366  days 
100  years 


1  hour. 

1  day. 

1  week. 

1  common  year. 

1  leap  year. 

1  century. 


1.  If  the  year  is  not  the  last  in  the  century,  its  number 
must  be  divisible  by  4  to  make  it  a  leap  year.  If  it  is  the 
closing  year  of  a  century,  it  is  not  a  leap  year  unless  its 
number  is  divisible  by  400. 


DENOMINATE  NUMBERS.  181 

2.  The  months  containing  30  days  are  April,  June,  Sep- 
tember, and  November. 

3.  The  months  containing  31  days  are  January,  March, 
May,  July,  August,  October,  and  December. 

4.  February  contains  28  days  in  a  common  year,  and  29 
days  in  a  leap  year. 

PROBLEMS. 

1.  How  many  seconds  are  there  in  a  day? 

2.  How  many  hours  are  there  in  a  common  year?  in  a 
leap  year? 

3.  Change  f  of  a  common  year  to  days,  hours,  minutes, 
and  seconds. 

4.  15  minutes  is  what  part  of  September? 

5.  Which  of  the  following  are  leap  years?  1866,  1880, 
1500,  2000,  1894,  1892. 

6.  Find  the  time  from  June  12,  1881,  to  March  5,  1884. 

Method.  Count  by  years  a.s  far  as  possible,  then  by  calendar  months, 
then  the  remaining  days;  thus,  from  June  12,  1881,  to  June  12,  1883,  is  2 
years.  From  June  12,  1883,  to  February  12,  1884,  is  8  months.  There 
are  1 7  days  left  in  February  and  5  in  March,  hence  the  time  is  2  years,  8 
months,  and  22  days. 

Note.  A  calendar  month  beginning  with  the  first  day  of  the  month 
completes  the  month ;  a  calendar  month  beginning  with  any  other  day 
ends  with  the  next  preceding  day  in  the  following  month.  The  periods 
Oct.  1-31,  June  10-July  9,  Jan.  31-Feb.  28,  are  calendar  months. 

7.  Find  the  time  : 

(1)  From  December  5,  1881,  to  June  16,  1886.- 
(2;  From  May  5,  1879,  to  September  20,  1883. 

(3)  From  August  16,  1888,  to  June  24,  1892. 

(4)  From  January  21,  1886,  to  November  7,  1893. 

8.  How  many  days  from  January  23  to  July  29,  common 
year? 

Note.     8 +  28 +  31,  etc. 

9.  How  many  days  from  October  16,   1890,  to  June  3, 


182  NEW  ADVANCED  ARITHMETIC. 

242.  THE  CALENDAR. 

1.  There  are  three  natural  time  uuits :  the  year,  the  month,  and  the 
day. 

2.  The  natural  day  from  midnight  till  midnight  is  not  of  uniform 
length.     The  mean  solar  day  is  the  average  of  all  the  days  in  the  year. 

3.  The  month,  the  period  from  one  new  moon  to  the  next,  equals  29h 
days  nearly. 

4.  The  year,  the  period  between  two  successive  vernal  equinoxes,  equals 
365.2422  days. 

5.  A  calendar  is  an  adjustment  of  tliese  natural  time  uuits  for  civil 
purposes.  A  lunar  calendar  makes  the  mouth  a  leading  luiit.  Mouths 
are  alternately  of  29  and  30  days.  The  year  of  12  months  or  354  days  is 
called  a  lunar  year.  Since  the  lunar  year  is  11  days  too  short,  extra 
mouths  (7  in  19  years)  must  be  added  from  time  to  time. 

6.  The  lunar  year  is  still  used  iu  Mohammedan  countries. 

7.  The  Juliau  calendar,  established  by  Julius  Ca;sar  in  46  b.  c,  pro- 
vided a  common  year  of  3G5  days  and,  every  fourth  year,  a  leap  year  of 
366  days.  The  mouths  beginning  with  Jlarch  were  alternately  of  31  aud 
30  days.  Later  August  was  given  an  additional  day  at  the  expense  ol 
February,  and  Octolier  and  December  were  made  months  of  31  days  instead 
of  September  aud  November. 

8.  The  average  Julian  year  of  365.25  days  exceeds  the  true  solar  year 
by  .0078  day  or  j^-g  day.  Gregory  XIII.  in  1582  corrected  as  much  of  tlie 
excess  as  had  accumulated  since  325  a.  d.  by  decreeing  that  the  5th  of 
October  should  be  the  15th.  He  provided  that  every  year  that  is  divisible 
by  4,  and  not  by  100,  is  a  leap  year.  Of  the  century  years,  only  those 
divisible  by  400  are  leap  years.  The  Gregorian  calendar  thus  provides  'j7 
leap  years  in  400  years. 

9.  The  Gregorian  calendar  was  not  adopted  by  the  nations  of  northern 
Europe  until  1700.  It  was  adopted  in  England  in  1752.  Russia  still  uses 
the  Julian  calendar. 

243.     Questions. 

1.  What  years  since  1582  have  been  leap  years  in  Russia  and  not  in 
Italy  1 

2.  Christmas  in  Russia  comes  how  many  days  later  than  with  us  ? 

3.  By  what  fraction  of  a  day  does  tlie  average  Gregorian  year  exceed 
the  true  year  1 

4.  In  how  many  years  will  the  excess  in  the  Gregorian  calendar  amount 
$0  one  day  1 

5.  wiiat  is  the  length  of  the  true  year  in  days,  hours,  minutes,  and 
seconds  7 


DENOMINATE  NUMBERS.  183 


244.     MISCELLANEOUS    TABLES. 

1,      12  oues    —  1  dozen. 
12  doz.    —  1  gross. 
12  gross  r=  1  great  gross. 
20  ones    =  1  score. 


2.     24  sheets  of  paper  ~  1  quire. 
20  quires  =  1  ream. 

2  reams  =  1  bundle. 

5  bundles  =  bale. 


Mariners'  Measure. 

6  feet  =  1  fathom. 

120  fathoms  =  1  cable-length, 

80  cable-lenoths  =  1  mile. 


245.     ADDITION. 

Define  addition,  sum.     Tell  how  the  numbers  are  written 
for  addition. 

1.    "^hat  is  the  sum  of  the  followino;  numbers : 


1  mi. 

1.S2 

rd. 

4 

yd. 

1 

ft. 

7  in. 

2 

309 

.") 

2 

9 

5 

169 

3 

0 

4 

8 

274 

4 

2 

11 

15 

0 

3 

2 

6 

2.  A  man  travelled  23  mi.  186  rd.  on  Monday.  19  mi. 
29.')  rd.  4  yd.  on  Tuesday,  36  mi.  83  rd.  5  yd.  on  Wednes- 
day, 19  mi.  317  rd.  2  yd.  on  Thursday,  28  mi.  297  rd.  on 
Friday,  and  34  mi.  168  rd.  on  Saturday;  how  far  did  he 
travel  in  the  six  days? 


184  NEW  ADVANCED  ARITHMETIC. 

3.    Add: 


4.    Add: 


5.   Add: 


6.    Add 


7.    Add; 


3  sq.  yd. 

5  sq.  ft. 

68  sq.  in 

. 

5 

7 

124 

7 

6 

99 

2 

8 

136 

4 

2 

79 

2  A. 

121  sq.  rd 

.  17  sq.  yd. 

3 

139 

24 

7 

86 

19 

12 

117 

28 

1 

110  sq.  rd. 

26  sq.  yd. 

4  sq.  ft. 

83  sq.  in.      j 

129 

14 

7 

132          1 

147 

29 

6 

59           ! 

153 

17 

2 

126 

88 

21 

8 

94          i 

2  cu.  yd. 

19  cu.  ft. 

824  cu 

.  in.            [ 

5 

24 

1232 

6 

16 

714 

12 

21 

936 

17 

26 

1532 

j 

2 

15 

1129 

1 

5  cd» 

6  cd.  ft.  14  cu. 

fto                 ! 

12 

4 

9 

6 

7 

12 

8 

2 

11 

27 

5 

15 

8.  Find  the  sum  of  5  cd.  7  cd.  ft.,   6  cd.  9  cd.  ft.,   12 
cd.  4  cd.  ft.,  9  cd.  4  cd.  ft. 

9.  Sold  3  cd.  5  cd.  ft.  12  cu.  ft  ,   12  cd.  14  cu.  ft.,  9  cd. 
5  cd.  ft.  15  cu.  ft.     How  much  in  all? 

10.  Find  the  sum  of  2  gal.  3  qt.  1  pt.,   5  gal.  2  qt    1  pt., 
7  gal.  1  qt.,  10  gal.  1  pt.,  6  gal.  2  qt.  1  pt. 

11.  Find  the  sum  of  4  bu.  3  pk.  5  qt.,  6  bu.  2  pk.  7  qt., 
12  bu.  1  pk.  6  qt. 


DENOMINATE  NUMBERS.  185 

12.  Add  2  lb.  8  oz.  14  pwt.  18  gr.,  3  lb.  7  oz.  12  pwt. 
10  gr.,  5  lb.  10  oz.  8  pwt.  17  gr.,  12  lb.  15  pwt.  21  gr. 
Troy. 

13.  Add  1  lb.  3  oz.  5  dr.  2  sc.  16  gr.,  2  lb.  7  oz.  6  dr. 
14  gr.,  8  oz.  2  sc.  18  gr.,  5  dr.  2  sc.  9  gr.  Apothecaries'. 

14.  Made  4  purchases  in  London,  costing  respectively 
£4  8  s.  7  d.,  £7  15  s.  9  d.,  £5  18  s.  8  d.,  and  £10  7  s.  6  d. 
What  was  the  amount  expended? 

15.  What  is  the  difference  in  latitude  of  Boston  (42°  21' 
24"  N.)  and  Rio  de  Janeiro  (22°  54'  S.)  ? 

16.  A  man  bought  a  quantity  of  broadcloth  for  £  17  9  s. ; 
of  silk  for  £23  11  d.;  of  cotton  goods  for  18  s.  9^  d. ;  of 
linen  goods  for  £29  15  s.  11|  d. ;  of  groceries  for  17  s. 
81  d. ;  of  boots  and  shoes  for  £31  19  s.  b\  d.  What  did 
he  pay  for  all? 

17.  Add  }  of  a  mile  and  f  of  8  rods. 

18.  Add  ^^  of  a  square  yard  and  .37  of  a  square  rod. 

19.  Add  2f  weeks,  5.33i  days,  6.375  hours. 

20.  Add  £  21,  f  of  £  1,  .27  of  £  1. 

246.     SUBTRACTION. 

Define  subtraction,  minuend,  subtrahend,  remainder.  Tell 
how  the  numbers  are  written  for  subtraction.  Give  the  rule. 
What  is  the  proof  ? 

1.  The  distance  from  A  to  B  is  12  mi.  83  rd.  3  yd.  2  ft., 
and  from  A  to  C  is  8  mi.  117  rd.  5  yd.  1  ft. ;  the  first  dis- 
tance is  how  much  greater  than  the  second  ? 

2.  From  4  mi.  68  rd.  2  yd.  1  ft.  4  in.  take  2  mi.  97  rd. 
4  yd.  2  ft.  9  in. 

3  From  27  sq.  yd.  6  sq.  ft.  83  sq.  in.  take  16  sq.  yd. 
8  sq.  ft.  141  sq.  in. 

4  From  5  A.  83  sq.  rd.  13  sq.  yd.  5  sq.  ft.  67  sq.  in.  take 
2  A.  98  sq.  rd.  29  sq.  yd.  7  sq.  ft.  110  sq.  in. 


186  NEW  ADVANCED  ARITHMETIC. 

5.  From  2  sec.  512  A.  73  sq.  rd.  take  1  sec.  538  A. 
95  sq.  rd.  26  sq.  yd. 

6.  Find  the  difference  of  the  following : 

£  17  lis.  7  d.  2  qr. 

12 15 V 3^ 

4  15  7  3 

7.  Find  the  difference  between  15  cu.  yd.  18  cu.  ft.  1276 
cu.  in.  and  7  cu.  yd.  23  cu.  ft.  and  1528  cu.  in. 

8.  Bought  600  lb.  of  sugar.  Sold  124  lb.  6  oz.,  73  lb. 
13  oz.,    48  1b.  9  oz.,    173  lb.  14  oz.     How  much  was  left? 

9.  Bought  624  ed.  of  wood.  Sold  75  cd.  7  cd.  ft.,  116 
cd.  M  cu.  ft.,  124  cd.  5  cd.  ft.  12  cu.  ft.,  283  cd.  4  cd.  ft. 
10  cu.  ft.     How  much  was  left? 

10.  What  is  the  difference  between  8  lb.  Apothecaries' 
weight,  and  5  lb.  7  oz.  4  dr.  2  sc.  16  gr.  ? 

11.  A  is  in  long.  124°  42'  36"  E.,  and  B  is  67°  49'  24"  E. 
What  is  their  difference  in  longitude? 

12.  From  a  cask  containing  38  gallons  the  following 
amounts  were  drawn  :  4  gal.  3  qt.  1  pt.,  7  gal.  2  qt.,  12  gal. 
1  qt.  1  pt.,  8  gal.  3  qt.  1  pt.  How  much  was  left  in  the 
cask  ? 

13.  From  I  of  1  rd.  take  .63  of  1  rd. 

14.  From  the  sum  of  -{'^  of  3  lb.  and  .35  of  2  lb.,  take 
the  sum  of  5  of  5  oz.  and  .64  of  2  lb.  Apothecaries'. 

15.  Started  to  walk  124  miles.  Went  the  first  day  18 
miles,  74  rods ;  the  second  day,  §  of  23  miles ;  the  third 
day  .28  of  95  miles.      What  distance  remained? 

16.  Find  difference  in  time  between  Januar}'  21,  1X95,  and 
July  28,  1848;  between  August  12,  1876,  and  May  10, 
1890. 

17.  From  12  lb.  9  oz.  7  pwt.  11  gr.  take  7  lb.  10  oz.  15 
pwt.  18  gr. 


DENOMINATE  NUMBERS.  187 

18.  How  long  a  time  from  the  battle  of  Bunker  Hill  to 
the  firing  on  Fort  Sumter? 

19.  A  cylindrical  cistern  is  10  feet  deep  and  has  a  diame- 
ter of  8  feet.  What  is  its  capacity  in  gallons?  (tt  =  2^.) 
In  barrels?  (3H  gallons.)  Being  |  full,  the  following 
amounts  were  drawn  out :  1^  barrels,  2\  barrels,  3.25  barrels, 
6.325  barrels.     How  many  barrels  were  left  in  the  cistern? 

1728  X  22  X  16  X  10  X  2 
Suggestion.      z „„,        „„ =  barrels. 

20.  From  I  of  1  cord  take  .16  of  3  cords. 

21.  A  man  paid  £11  12  ?.  8^  d.  for  a  wagon.  He  gave 
the  merchant  a  £  20  note.  What  change  should  he  receive  ? 
Give  the  rule  for  subtraction  of  simple  numbers. 

22.  A  man  having  34  cords  of  wood  sold  to  one  man  5 
cd.  7  cd.  ft.  12  cu.  ft.,  to  another  15  cd.  14  cu.  ft.,  and  to  a 
third  8  cd.  5  cd.  ft.     How  much  had  he  left? 

247.     Multiplication. 

Define  multiplication,  multiplicand,  multiplier,  product. 
Give  the  rule  for  multiplication  of  simple  numbers.  AVhat 
is  the  denomination  of  the  product?  Be  able  to  give  the 
analysis  of  reduction  at  each  step. 

1.    Multiply  £7  12  s.  9  d.  3  qr.  by  9. 

FOKM. 

£7  12s.  9d.  3qr. 

9 


63  1U8  81  4)27(6 

5  7  6  24 

68        20)115(5  12)~sf(7  3 

100  84 

15  ~3~ 

2.  Multiply  3  mi.  25  rd.  4  yd.  2  ft.  8  in.  by  12. 

3.  Multiply  8  cu.  yd.  13  cu.  ft.  124  cu.  in.  by  24. 


188  NEW  ADVANCED  ARITHMETIC. 

4.  A  pile  of  wood  is  4  feet  wide,  4  feet  high,  and  27  feet 
long.  How  many  cords  in  15  S'leh  piles?  What  is  it  worth 
at  84.50  a  cord? 

5.  A  ship  sails  from  N.  Y.,  'ongitude  74°  0'  3"  W.,  and 
makes  an  average  dail}^  easting  of  'J°  24'  3G."  What  is  her 
longitude  at  the  end  of  7  days? 

6.  27  cans  hold  an  average  of  10  gal.  3  qt.  1  pt.  How 
much  do  they  all  contain? 

7.  A  bin  is  8  feet  wide,  12  feet  long,  and  7  feet  high. 
How  many  bushels  of  shelled  corn  will  it  hold  ?  How  much 
will  18  such  bins  hold? 

8.  36  men  worked  an  average  of  ''2  d.  7  hr.  and  30  min. 
How  much  money  will  pay  them,  at  81.25  a  day,  counting  10 
hours  to  the  day? 

9.  Find  the  cost  to  the  druggist  of  36  prescriptions  of 
quinine,  each  containing  24  grains,  if  quinine  cost  42  cents 
for  an  avoirdupois  ounce  (437i  grains). 

10.  Sold  28  loads  of  oats,  each  containing  74  bu.  3  pk., 
at  IGj  cents  a  bushel.     What  was  the  amount  received? 

11.  Bought  a  city  lot  containing  |  of  an  acre  at  82.25  a 
square  yard.     What  did  it  cost? 

12.  Multiply  15  hr.  24  min.  38  sec.  by  42. 

13.  Multiply  16°  17'  22"  by  76. 

14.  Multiply  128  lbs.  7  oz.  by  56. 

Division. 

248.  Define  partition,  divisor,  dividend,  quotient,  re- 
m.ainder.  What  terms  are  alike  ?  What  kind  of  a  number 
is  ihe  di\T[sor? 

249.  Define  measurement,  divisor,  dividend,  quotient,  re- 
mainder. Which  terms  are  alike?  What  kind  of  a  number 
is  the  quotient? 


DENOMINATE  NUMBERS.  189 

1.  If  £18  12  s.  3  d.  3  qr.  be  divided  equally  among  7 
persons,  what  will  each  receive? 

FORM. 

£  s.  d.  qr. 

7)  18  (£2  12  3  3 

14  80  12  4 

~T  7)'92"(13s.         7)'T5~(2d.         lyTilc^x. 

91  U  7 

2.  How  many  articles  at  £2  7  s.  8  d.  3  qr.  each  can  be 
purchased  for  £40  11  s.  4  d.  3  qr.? 

Analysis.  To  simplify  this  problem,  divisor  and  dividend  should  be 
reduced  to  the  lowest  deuoniiuation  found  in  the  numbers.  The  divisor 
equals  2291  qr.  The  dividend  equals  38947  qr.  There  are  17  229rs  in 
38947;  hence,  17  such  articles  can  be  purchased. 

3.  Divide  8  mi.  186  rd.  4  3'd.  by  7. 

4.  Divide  8  mi.  186  rd.  4  3'd.  by  7  3'd. 

5.  If  a  field  containing  71  A.  82  sq.  rd.  be  divided  into  8 
equal  parts,  what  will  each  part  contain  ? 

6.  Bought  36  2  X  4  16-foot  studs,  48  2  X  8  18-foot  joists, 
4  sills  8  X  8  16  feet  long,  at  $18.50  per  thousand;  1,250  feet 
flooring  at  $32;  1,460  feet  sheathing  at  820,  and  1,480  feet 
siding  at  833.50.  If  the  bill  were  divided  into  4  equal  pay- 
ments, what  would  each  amount  to? 

7.  How  many  paving-stones  4  ft.  4  in.  long  and  3  ft. 
wide  will  be  needed  to  make  a  3-foot  walk  186  ft.  8  in.  long? 

8.  How  many  bricks,  of  ordinary  size,  will  be  required  to 
pave  a  court  16  feet  wide  and  80  ft.  8  in.  long? 

9.  DiWde  4  lb.  2  oz.  13  pwt.  by  1  lb.  4  oz.  17  pwt.  16  gr. 

10.  Divide  £19  17  s.  Od.  2  qr.  by  £1  11  d. 

11.  If  a  man  travel  at  an  average  rate  of  4  mi.  25  rd.  5 
yd.  an  hour,  how  many  hours  wiU  be  required  to  travel  175 
miles  ? 


190  NEW  ADVANCED  ARITHMETIC. 

12  Divide  24  T.  826  lbs.  by  724  lbs.  8  oz. 

13  Bought  15  cd.  7  cd.  ft.  12  cu.  ft,  which  was  dehvered 
in  16  loads.     What  was  the  average  load? 

14.  If  7  hr.  15  mill.  40  sec.  is  the  average  time  required 
for  a  man  to  produce  a  certain  article,  how  many  such  arti- 
cles can  be  produced  in  348  working  hours? 

15.  How  many  cases,  each  holding  2  gal.  3  qt.  1  pt.,  will 
be  needed  to  hold  48  gal.  3  qt.  1  pt.  ? 

16.  A  box  has  a  capacity  of  1  bu.  3  pk.  5  qt.  How  many 
times  must  a  laborer  fill  it  to  remove  324  bu.  2  pk.  of  oats? 

17.  A  cellar  is  18  ft.  6  in.  by  24  ft.  4  in.  and  5  ft.  deep. 
How  many  loads  J  of  a  cu.  yd.  each  will  the  excavated  dirt 
make? 

18.  A  man,  having  44  A.  96  sq.  rd.  of  land,  sold  5  A.  92 
sq.  rd.     What  part  of  the  land  does  he  still  own? 

19.  A  cubical  tank,  10  feet  square  at  the  base,  has  a 
capacity  of  8,000  gallons.     What  is  its  height? 

20.  A  cubic  foot  of  pure  gold  may  be  coined  into  how 
many  dollars?     Gold  is  19]  times  as  heavy  as  water. 

250.     MISCELLANEOUS   PROBLEMS, 
REVIEW. 

1.  Find  the  sum  of  $83.2,  $632.7,  $504.9,  $473.3,  $712.5, 
S  190.04. 

2.  Find  the  sum  of  $6041.072,  $4003.926,  $9621.863, 
$7028.414,  $8631.372,  $36027.496,  $48971.022. 

3.  What  is  the  cost  of  7  articles  at  $8,464  each? 
Of  36  articles  at  $15,842  each? 

Of  329  articles  at  $76,575  each? 
Of  974  articles  at  $83,125  each? 
Of  87  articles  at  $479,375  each? 

4.  If  23  barrels  of  flour  cost  $155.25,  what  is  the  price 
per  barrel? 


.       DENOMINATE  NUMBERS.  191 

5.  If  725  acres  of  land  cost  $49,571,875,  -what  is  the  price 
per  acre  ? 

6.  What  is  the  difference  between  87000  and  82874.6G4? 

7.  A  man  received  86,126.82  for  his  farm,  82,579.12  for 
his  stock,  and  81,966.47  for  his  grain.  He  bought  a  house 
for  83,582.96;  furuitui-e  for  81,391.65  ;  a  horse  for  8164.25  ; 
a  carriage  for  8164.28 ;  and  harness  for  836.80.  How  much 
money  did  he  have  left? 

8.  A  man  purchased  a  library  for  87 63. 65 J,  paj'ing  an 
average  price  of  82.34^  per  volume.  How  many  volumes 
did  he  buy? 

9.  Find  the  entire  cost  of  the  following  articles :  1  desk, 
S28.50;  1  bookcase,  868.30;  1  half  dozen  chairs,  818.25; 
1  rocker,  812.70;  1  bedstead,  829.50;  1  bureau,  829.58; 
I  washstand,  811. "6;  one  stove,  837.49;  1  table,  824.76; 
1  lounge,  819.46. 

10.  At  47:^  cents  each,  how  many  bushels  of  corn  cau  be 
purchased  for  8343.98? 

11.  A  paid  8491.76  for  a  pair  of  horses,  and  8278.97  for 
a  carriage.  How  much  more  did  he  pay  for  the  horses  than 
tor  the  carriage? 

12.  86215.824  is  how  much  more  than  81987.948? 

13.  A  street-car  company  bought  864  mules,  paying 
$79,200  for  them;  what  was  the  average  price? 

14.  At  817. 58^-  each,  how  many  calves  can  be  bought  for 
85,697? 

15.  A  man  bought  six  farms.  For  the  first  he  paid 
86,012.07;  for  the  second,  84,631.26;  for  the  third, 
$3,712.84  ;  for  the  fourth,  88,067.53  ;  for  the  fifth, 
87,824.86;  for  the  sixth,  6,098.94.  What  did  he  pav  for 
aU? 

16.  A  merchant  sold  a  man  the  following  articles  :  sugar, 
$1.40;  coffee,  97  cents;  tea,  83  cents;  salt,  48  cents;  flour, 
$1.85;    apples,   82.38;    potatoes,   86   cents;    molasses,   85 


192  NEW  ADVANCED  ARITHMETIC. 

cents.     He  received  in  payment  a  820-bill.     "What  amount 
of  money  should  he  return? 

17.  What  is  the  entire  cost  of  the  following  articles :  one 
horse,  8116.87;  one  buggy,  Si 29. 40;  a  set  of  harness,  828.90; 
a  whip,  §2.55;  one  wagon,  865.75;  one  blanket,  83.78;  one 
sleigh,  836.47? 

18.  Change  873  yards  to  a  compound  number. 

19.  Change  2  ft.  3  in.  to  a  fraction  of  a  mile. 

20.  Add :  4  bu.  3  pk.  5  qt.  1  pt. ;  6  bu.  2  pk.  7  qt. ;  12  bu. 
1  pk.  6  qt.  1  pt. ;  and  23  bu.  3  qt. 

21.  A  railway  train,  running  at  the  average  rate- of  34  mi. 
68  rd.  4  yd.  2  ft.  per  hour,  went  from  A  to  B  in  9  hours. 
What  is  the  distance  between  the  two  places? 

22.  Multiply  217  rd.  4  yd.  2  ft.  7  in.  by  23. 

23.  The  area  of  a  floor  is  25  sq.  yd.  6  sq.  ft.  83  sq.  in. 
What  is  the  entire  area  of  1 2  such  floors  ? 

24.  How  much  laud  is  there  in  9  fields,  if  each  contain 
53  A.  47  sq.  rd.  26  sq.  yd.  ? 

25.  How  many  4-ouuce  vials  can  be  filled  from  5  gal.  3  qt. 
1  pt.  3  gi.  of  alcohol? 

26.  Divide  583  bu.  3  pk.  7  qt.  of  corn  into  16  equal  parts. 

27.  How  many  40-gallon  barrels  of  water  will  a  cubical 
cistern  contain  that  is  10  feet  deep? 

28.  Multiply  45  A.  24  sq,  rd.  18  sq.  yd.  by  38. 

29.  37  equal  quantities  of  land  contain  37  sec.  201  A. 
88  sq.  rd.  23  sq.  yd.  2  sq.  ft.  72  sq.  in.  What  does  each 
contain? 

30.  A  railway  train,  moving  at  a  uniform  rate,  ran  307 
mL  299  rd.  3|^  yd.  in  9  hours.     What  was  the  rate  per  hour? 

31.  How  many  revolutions  will  a  carriage  wheel,  whose 
circumference  is  11  ft.  4  in.,  make  in  describing  a  distance 
of  1  mi.  125  rd.  4  yd.  10  in.? 

32.  Divide  41  rd.  4  yd.  10  in.  by  4  yd.  2  ft.  8  in. 


DENOMINATE  NUMBERS.       .  193 

33.  If  80  cu.  3'd.  4  cu.  ft.  848  cu.  in.  of  earth  were  re- 
moved iu  28  equal  loads,  how  much  did  each  load  contain? 

S*.  How  many  piles  of  wood,  each  containing  2  cd.  75  cu. 
ft.,  can  be  made  from  93  cd.  12  cu.  ft.  ? 

35.  3.46  miles  =  ? 

36.  Change  4  inches  to  the  decimal  of  a  mile. 

37.  3  yd    1  ft.  8  in.  is  what  part  of  225  rd.  4  yd.? 

38.  What  is  the  sum  of  the  following  numbers? 


1  mi. 

182  rd. 

4 

yd. 

1  ft. 

7  in. 

2 

309 

5 

2 

9 

5 

169 

3 

0 

4 

8 

274 

2 

2 

11 

15 

0 

1 

2 

6 

39.  Divide  66  sq.  yd.  6  sq.  ft.  by  2  sq.  yd.  7  sq.  ft. 

40.  How  many  lots,  each  containing  4  A.  36  sq.  rd.,  can 
be  formed  from  50  A.  112  sq.  rd.  ? 

41.  From  a  40-gal.  barrel  of  vinegar  a  merchant  sold  to 
one  man  4  gal.  3  qt.  1  pt.  1  gi.  ;  to  a  second  5  gal.  2  qt.  3  gi. ; 
and  to  a  third  13  gal.  1  qt.  1  pt.  2  gi.  What  amount  was 
left  in  the  barrel? 

42.  What  quantity  of  oats  will  15  bins  contain  if  the 
capacitj^  of  each  be  186  bu,  3  pk.  7  qt.  ? 

43.  Change  2  yd.  1  ft.  11  in.  to  a  fraction  of  a  rod. 

44.  4  of  a  mile  =  ? 

45.  §  of  a  rod  =  ? 

46.  How  much  wood  is  there  in  24  piles  of  wood,  each 
containing  9  cd.  86  cu.  ft.? 

47.  Multiply  18  cu.  ft.  724  cu.  in.  by  46. 

48.  Reduce  2  mi.  180  rd.  3  yd.  2  ft.  10  in.  to  inches. 

49.  How  many  feet  in  321  rd.  4  yd.  1  ft.? 

50.  Reduce  87889  inches  to  integers  of  higher  denomi- 
nations. 


194  NEW  ADVANCED  ARITHMETIC. 

51.  Reduce  i\  of  a  square  mile  to  integers  of  lower 
denominations. 

52.  Reduce  .028  of  an  acre. 

53.  108  square  inches  is  what  part  of  a  square  rod? 

54.  2  sq.  yd.  5  sq.  ft.  64  sq.  in.  is  what  part  of  16  sq. 
yd.  2  sq.  ft.  76  sq.  in.?  Express  the  result  as  a  decimal 
fraction. 

55.  Reduce  4813  feet  to  integers  of  higher  denominations. 

56.  Change  429  yards  to  a  compound  number. 

57.  An  English  officer  bought  65  horses  for  his  company 
at  an  average  price  of  £21  12  s.  6  d. ;  what  was  the  aggre- 
gate cost? 

58.  How  many  bushels  of  oats  will  a  rectangular  bin  con- 
tain that  is  6  ft.  long,  4  ft.  wide,  and  5  ft.  8  in.  high? 

59.  A  farmer  bought  the  following  tracts  of  land,  all  lying 
in  the  same  section  : 

The  N.  J-  of  the  S.  E.  I  of  the  S.  W.  -}. 
The  S.  J-  of  the  N.  E.  ^  of  the  S.  W.  i. 
The  N.  J  of  the  S.  W.  i  of  the  S.  E.  i. 
Draw  a  diagram  of  the  section  and  show  his  purchase. 
What  did  it  cost  at  $62.50  an  acre? 

60.  Find  the  cost  of  the  posts  and  fencing  necessary  to 
build  a  four-board  fence  around  the  land  just  described : 
posts  being  worth  22  cents  each,  and  placed  8  feet  apart, 
and  the  fencing  being  1  in.  thick,  6  in.  wide,  16  ft.  long,  and 
costing  $18.25  per  M.? 

61.  Reduce  }^  of  an  acre  to  square  rods,  etc. 

62.  2^  qr.  is  what  part  of  £l  ? 

63.  How  many  revolutions  will  a  carriage  wheel,  whose 
circumference  is  14  ft.  8  in.,  make  in  going  2  mi.  84  rd.? 

64.  Change  6  grains  Troy  to  the  decimal  of  a  pound. 

65.  What  are  7  loads  of  hay,  each  weighing  2,460  pounds, 
worth  at  S8.25  a  ton? 


DENOMINATE  NUMBERS.  195 

66.  Add  §  ci  an  acre,  5  of  a  square  rod,  |  of  a  square 
yard,  and  -^^  cf  a  square  foot. 

67.  What  is  the  value  of  3  oz.  5  dr.  of  quinine  at  3  cents 
ji  grain? 

68.  Reduce  /^  of  a  common  year  to  days,  hours,  etc. 

69.  A  man  sets  his  watch  at  Chicago  local  time.  After 
travelling  for  some  tune  he  finds  that  it  is  1  hr.  24  min.  30 
sec.  faster  than  the  local  time  where  he  is.  What  is  his 
longitude  ? 

70.  12"  of  arc  is  what  part  of  a  circumference? 

71.  What  is  the  cost  of  the  following  piles  of  cord- wood 
at  $4.75  a  cord? 

1  pile,  8  ft.  high,  22  ft.  long. 

2  piles,  each  6^  ft.  high,  31  ft.  long. 
1  pile,  9^  ft.  high  and  32 i  ft.  long. 

72.  A  square  cistern,  whose  bottom  is  8  feet  on  a  side,  is 
12  feet  deep.  How  many  gallons  of  water  are  there  in  it 
when  it  is  I  full? 

73.  Change  \\  oi  o.  cubic  yard  to  cubic  feet  and  cubic 
inches. 

74.  What  is  the  cost  of  the  Brussels  carpet,  at  Si. 6 3  a 
yard,  to  cover  the  floor  of  a  room  that  is  22  feet  long  and  19 
feet  wide,  the  strips  to  run  the  long  way? 

75.  What  change  shall  be  made  in  a  watch  that  is  set  to 
New  York  local  tune  to  make  it  agree  with  Chicago  local 
time? 

76.  When  eggs  are  sold  at  the  rate  of  2  for  3|  cents, 
wh.at  is  the  cost  >f  4^  gross? 

77.  Find  the  cost  of  the  following  bill  of  lumber  at 
$21.50  per  M.  ; 

5  sills,  8  X  10,  18  ft.  long. 
36  joists,  2  X  10,  16  ft.  long. 
42  studs,  2x4,  22  ft.  long. 
70  boards,  averaging  9  in.  wide  and  14  ft.  long. 


196  NEW  ADVANCED  ARITHMETIC. 

78.  If  17  T.  15  cwt.  3  qr.  12  lb.  of  coal  be  divided  equally 
among  12  bins,  how  much  will  each  contain? 

79.  How  many  quarts  of  milk  will  a  vessel  hold  whose 
capacity  is  1  peck? 

80.  Add  .44  of  a  common  year  to  29  days,  19  hr.  18  min. 

81.  Divide  an  angle  of  100°  10'  1"  into  13  equal  parts. 

82.  Reduce  8975  grains  Troy  to  a  compound  number. 

83.  Reduce  2  lb., 9  oz.  5  dr.  2  sc.  15  gr.  Apothecaries'  to 
grains. 

84.  A  room  is  15  feet  by  20  feet  wjth  walls  12  feet  high. 
There  are  3  windows  2 J  feet  by  6  feet,  and  2  doors  2  feet 
8  inches  by  8  feet  4  inches.  If  wall-paper  cost  22  cents  a 
roll,  and  border  5  cents  a  yard,  what  is  the  whole  cost  for 
walls  and  ceiling? 

85-    Make  a  bill  of  the  following  items,  and  receipt  it : 
32  pounds  of  sugar,  at  6^-  cents. 
48  yards  of  calico,  at  8^  cents. 
28  bushels  of  potatoes,  at  37^  cents. 
18  bushels  of  apples,  at  87J  cents. 
32  yards  of  cloth,  at  75  cents. 
3  dozen  plates,  at  50  cents. 

86.  What  is  the  cost  of  3^  reams  of  letter-paper,  at  12^ 
cents  a  quire  ? 

87.  "What  is  the  time  from  March  11,  1883,  to  January 
19,  1889? 

88.  Change  88537  square  feet  to  square  yards,  etc. 

89.  How  many  days  from  January  25,  1892,  to  December 
11? 

90.  A  steamboat  going  down  stream  is  propelled  12  miles 
an  hour  by  steam,  and  320  feet  a  minute  by  the  current ;  in 
what  time  can  she  go  175  miles?  In  what  time  wUl  she  go 
the  same  distance  up  stream? 

91.  Reduce  25"  to  the  decimal  of  a  degree. 


DENOMINATE  NUMBERS.  197 

92.  To  ^  of  a  mile  add  7569  inches,  and  multiply  the 
result  by  7. 

93.  From  8.36  bushels  take  3f  pecks,  and  divide  the 
remainder  by  12. 

94.  2\  quarts  is  what  part  of  3  gal.  1  pt? 

95.  1  gill  is  what  part  of  2  gallons?  Change  the  result 
to  a  decimal  fraction. 

96.  In  a  pacing  race  two  horses  started  together  and 
went  a  mile.  The  time  of  the  faster  was  2  min.  6  sec,  and 
of  the  other  2  min.  6|  sec.  How  much  was  the  winning 
horse  ahead  at  the  finish? 

97.  When  a  locomotive,  having  a  driving  wheel  5  feet  and 
10  inches  in  diameter,  is  running  at  the  rate  of  a  mile  in  a 
minute,  how  many  revolutions  do  the  "  drivers"  make  in  a 
second  ? 

98.  Sound  travels  at  the  rate  of  1,120  feet  a  second  under 
ordinary  conditions.  If  the  report  of  a  gun  is  heard  3|^ 
seconds  after  the  flash  of  the  discharge  is  seen,  what  part  of 
a  mile  is  the  observer  from  the  gun  ? 

99.  Over  what  area  may  a  horse  graze  if  tied  to  a  stake 
by  a  50-foot  rope  ? 

100.  Over  what  area  may  he  graze  if  tied  to  the  corner  of 
a  barn  30'  X  40'  by  a  50-foot  rope?     Draw  diagram. 

101.  An  experiment  showed  that  a  current  of  electricity 
passed  over  7,200  miles  in  f  of  a  second.  In  how  long  a 
time  would  it  describe  the  equatorial  circuit  of  the    earth? 

102.  Out  of  a  sheet  of  paper  8"  X  10"  cut  a  circle  6 
inches  in  diameter.     What  part  of  the  paper  is  cut  away  ? 

103.  What  is  the  longitude  of  Quebec  if  it  is  5  minutes 
and  42  seconds  past  1  p.  m.,  when  it  is  noon  at  Chicago? 

104.  Water  flows  into  a  tank  through  three  pipes.  The 
first  would  fill  it  in  3J  hours,  the  second  in  4i  hours,  and  the 
third  in  5^  hours ;  in  what  time  will  the  tliree  pipes  fill  it  ? 

14A 


198  NEW  ADVANCED  ARITHMETIC. 

105.  How  many  2-iuch  circles  equal  in  area  1  6-inch 
circle  ? 

106,  A  borrower  paid  8347  for  the  use  of  $2,400,  paying 
7%  of  the  amount  loaned  for  its  use  for  a  year.  How  many 
years,  months,  and  days  should  he  keep  it,  counting  30  days 
for  a  month? 

-  107  A  52-gallon  oil  barrel  was  J  full.  13  gallons  were 
drawn  out.  What  fraction  of  its  capacity  did  it  then  con- 
tain?    Change  this  fraction  to  a  decimal. 

108.  At  the  time  of  her  marriage,  8  years  ago,  Mrs.  S. 
was  ten  years  younger  thfin  her  husband.  Her  age  is  now 
J  of  his.     What  was  her  age  at  the  time  of  her  marriage? 

109.  In  a  school  of  57.5  pupils  the  number  of  boys  is  /^ 
of  the  number  of  girls.     How  many  are  there  of  each? 

110.  What  is  the  cost  of  a  city  lot  80'  X  IGO'  if  sold  for 
as  many  silver  dollars  as  can  be  laid  upon  it  in  a  single  layer 
and  placed  side  by  side  in  rows  parallel  to  the  sides  of  the 
lot? 

111.  What  is  the  area  of  a  4-inch  circle?  of  an  8-inch 
circle?  Divide  the  latter  area  by  the  former.  The  first  is 
what  part  of  the  second? 

112.  What  is  the  area  of  a  5-inch  circle?  of  a  10-inch 
circle?     The  second  is  how  many  times  the  first? 

113.  If  the  radius  (R)  of  one  cu'cle  is  twice  the  radius  (r) 
of  a  smaller  circle,  ttR-  is  how  many  times  7:r'-? 

114.  How  large  a  water-pipe  is  needed  to  carry  4  times  as 
much  water  as  a  o-iuch  pipe  ? 

115.  What  is  the  cost  of  the  gold  in  a  14-carat  chain 
weighing  12  pennyweight  at  820.67  per  ounce? 

116.  A  cubic  inch  of  gold  has  been  beaten  so  thin  as  to 
t'over  ^'4  of  an  acre,      \^'hat  was  its  thickness? 

117.  How  many  layers  of  such  gold  would  o(iual  the  thick- 
ness of  a  leaf  of  this  book? 


DENOMINATE  NUMBERS.  199 

Note.  Measure  thickness  of  the  book  exclusive  of  covers,  and  divide 
by  the  number  of  leaves. 

118.  Simplify:  ^^  -  H  +  3} 

119.  An  iron  beam  16  ft.  loag,  2\  in.  wide,  and  8  in. 
deep  weighs  900  lbs.  This  specimen  of  irou  is  how  many 
times  as  heavy  as  an  equal  volume  of  water? 

120.  What  part  of  §  of  %  is  |  of  -^-?  Change  the  result  to 
a  decimal  fraction. 

121.  Shingles  are  sold  in  bundles,  each  containing  the 
equivalent  of  250  shingles  4  inches  wide.  If  shingles  are 
laid  4i  inches  to  the  weather,  how  many  bunches  must  be 
bought  for  a  roof  20'  X  30'?  What  is  the  cost  of  laying 
them  at  80  cents  per  square  ? 

Note.     A  "  square"  contains  100  square  feet. 

122.  Add  f  of  an  acre  and  217^  sq.  rd. 

123.  A  rug  16'  X  12'  is  placed  in  the  middle  of  a  floor 
19'  X  15'.  AVhat  is  the  width  and  area  of  the  uncovered 
strip  ? 

124.  Change  the  following  to  decimal  fractions  of  five 
places:    ^\,  yf^,  ^\.. 

125.  1,000,000  American  eagles  will  coin  into  how  many 
sovereigns? 

126.  An  English  immigrant  changes  his  money,  £248, 
into  federal  money.     What  does  he  receive  for  it? 

127.  Gold  is  19;^  times  as  heavy  as  water.  What  is  the 
weight  of  a  cubic  foot?     What  is  its  value? 

128.  If  the  top  of  3'our  desk  were  covered  with  gold  two 
feet  deep,  what  would  it  weigh?     What  would  be  its  value? 

129.  How  high  is  a  rectangular  block  of  gold  one  foot 
square  at  the  base  and  weighing  one  ton  ?    What  is  its  value  ? 

130.  The  length  of  one  degree  of  longitude  at  the  40th 
parallel  is  53.063  miles.  How  far  apart  do  two  men  live  od 
this  parallel  whose  noons  are  just  one  minute  apart? 


200  NEW  ADVANCED  ARITHMETIC. 

131.  How  long  is  the  40th  parallel? 

132.  The  length  of  a  degree  of  longitude  at  latitude  45  ^ 
is  48.995  miles.  Calculate  as  accurately  as  you  can  the  dis- 
tance on  this  parallel  from  the  Connecticut  River  to  St.  Paul, 
Minn. 

133.  Add  86512,  43972,  64829,  93517,  48695,  82724, 
60982,  93728,  46479,  794736,  and  get  a  correct  result  in  20 
seconds  or  less. 

134.  Multiply  96534  by  48967,  and  obtain  a  correct  result 
in  50  seconds  or  less. 

135.  Divide  8497068314  by  59637,  and  obtain  a  correct 
result  in  80  seconds  or  less. 

136.  Write  the  rule  for  the  multiplication  of  a  fraction, 
and  illustrate  it  by  an  example,  telling  how  you  know  that 
you  have  a  correct  result. 

137.  Write  the  rule  for  the  division  of  a  fraction  by  a 
fraction,  illustrate  it  by  an  example,  and  explain  in  writing 
how  you  know  that  you  have  a  correct  result. 

138.  If  8  men  can  do  a  piece  of  work  in  12 J  days  of  8 
hours  each,  in  how  many  10-hour  days  can  5  men  do  the 
same  work? 

The  following  27  problems  were  used  in  examinations  for 
State  certificates  in  Indiana  and  Illinois  : 

139.  At  90  cents  per  yard  how  much  will  it  cost  to  carpet 
a  room  20  by  27  feet  with  carpet  2^  feet  wide,  allowing  one 
foot  waste  on  each  cut  for  matching? 

140.  If  12i  yards  of  dress  goods  will  make  a  dress,  how 
many  yards  of  cambric  If  yards  wide  will  be  required  to  line 
one  half  of  it?     If  the  goods  are  1  yard  wide  ? 

141.  If  one  bushel  of  wheat  will  make  40  pounds  of  Hour, 
how  many  barrels  of  flour  can  be  made  from  the  contents  of 
a  bin  full  of  wheat,  the  dimensions  of  the  bin  being  10'  X 
5'  X  4'? 


DENOMINATE  NUMBERS.  201 

142.  A  can  do  a  piece  of  -svork  in  8^  hours,  A  aud  B 
together  can  do  it  in  4-^  hours,  and  A  and  C  can  do  it 
together  in  4  hours.  How  many  hours  will  it  take  B  and 
C  to  do  the  work? 

143.  How  much  will  it  cost  to  plaster  the  walls  and  ceiling 
of  a  room  27  ft.  long,  15  ft.  wide,  and  12  ft.  high,  at  25 
cents  a  square  yard,  allowing  432  sq.  ft.  for  doors  and 
windows? 

144.  Find  the  circumference  and  area  of  a  circle  whose 
diameter  is  2  ft.  4  in. 

145.  Divide  125.37  by  15.75.  Solve  by  analysis,  and 
show  why  the  rule  for  pointing  is  correct. 

146.  A  vessel  sailed  from  a  port  directly  on  a  line  of  lati- 
tude a  certain  distance,  then  sailed  due  north  a  certain  other 
distance,  when  the  captain  found  his  chronometer  forty  min- 
utes slow.  In  what  direction  had  he  first  sailed  and  how 
many  degrees? 

147.  A  gold  mine  produces  $420,000  in  a  single  year. 
How  many  pounds  av.  did  the  output  weigh,  23.22  grains  be- 
ino-  worth  81?     Its  volume? 


148.    Find  the  value  of     1  —  -x-z  +  o,  y    -    - ,  • 

149.'  Five  men  in  a  factory  accomplish  as  much  as  eight 
boys.  What  part  of  a  man's  work  does  a  boy  do?  Change 
this  result  to  per  cent.  What  per  cent  of  a  boy's  work  does 
a  man  do  ? 

150.  The  diameter  of  a  cylindrical  tank  is  10^  feet,  and 
its  length  is  30^  feet.     How  many  gallons  will  it  hold? 

151.  (4.4  —  .00027)  X  2.1  X  .005  -^  .000005. 

152.  After  paying  §  of  a  debt  aud  I  of  the  remainder,  I 
owe  S430.371  less  than  at  fii-st.  What  was  the  debt  at 
first? 

153.  Reduce  57  A.  96  sq.  rd.  to  the  decimal  of  a  square 
mile. 


202  NEW   ADVAXCED   ARITHMETIC. 

154.  A  man  walks  a  certain  distance  at  the  rate  of  4^ 
miles  an  hour,  and  rides  back  at  the  rate  of  7.^  miles  an 
hour.  If  it  takes  him  8  hours  to  go  both  ways,  what  is  the 
distance  ? 

155.  Find  the  cost  of  25  pieces  of  scantling  b"  X  3^",  10 
ft.  long,  at  $10.25  i?er  M. 

156.  When  it  is  4  hr.  20  min.  p.  m.  ,  65°  25'  west  longi- 
tude, what  is  the  time  17°  20'  east  longitude? 

157.  The  Capitol  at  Washington  is  751  feet  long  and  384 
feet  wide.     How  many  acres  does  it  cover? 

158.  If  it  cost  $120  to  build  a  wall  40  ft.  long,  14  ft. 
high,  1  ft.  6  in.  thick,  what  will  it  cost,  at  the  same  rate,  to 
build  a  wall  180  ft.  long,  21  ft.  high,  and  1  ft.  3  in.  thick? 

159.  A  lake  whose  area  is  45  acres  is  covered  with  ice 
an  inch  thick.  Find  the  weight  of  the  ice  in  tons,  if  a  cubic 
foot  weighs  920  ounces  avoirdupois. 

160.  A  can  hoe  a  row  of  corn  in  a  certain  field  in  30  min- 
utes, B  in  20  minutes,  and  C  in  35  minutes.  What  is  the 
least  number  of  rows  that  each  can  hoe  that  all  may  finish  at 
the  same  time? 

161.  A  owns  y\  of  a  ship's  cargo,  valued  at  $493.000 ;  B 
owns  ^1  of  the  remainder ;  C  owns  y\  as  much  as  A  and  B, 
and  D  owns  the  remainder.     How  much  does  each  own? 

162.  How  many  square  rods  in  a  piece  of  laud  %  of  a  mile 
long  and  ^  of  a  mile  wide? 

163.  Light  occupies  16  minutes  and  36  seconds  in  cross- 
ing the  earth's  orbit.  If  the  earth  is  95  millions  of  miles 
from  the  sun  what  is  the  velocity  of  light? 

164.  .0001  -^  .00000001  =? 

165.  A  man  bought  a  horse  and  a  carriage  for  $280.  § 
of  the  cost  of  the  carriage  was  3  of  the  cost  of  the  horse. 
What  was  the  cost  of  each? 


PER  CENT  A  GE.  20,'^ 


j^art  II. 

SECTION    YII. 

251.     PERCENTAGE. 

1.  Percentage  is  a  system  of  calculations  by  hundredths. 

2.  Per  cent  means  hundredth  or  hundredths.     1  per  cent 
is  iJ-^;  7  per  cent  is  -i^^;  f  per  cent  is  f  of  ^l^. 

3.  Any  per  cent  is  a  decimal   fraction  having   100  for  its 

denominator.     It  may  be  expressed  in  the  form  of  a  com- 

7        ^       81  . 
mon  fraction,  as  j^^'  j^'  j^'    in  the  form  of  a  decimal 

fraction,  as  .07,  .00§,  .08^,  or  with  the  per  cent  symbol,  as 
7%,  1%,  8^-%. 

4.  Per  cent  differs  from  decimal  fractions  in  general  iu 
two  ways : 

(1)  Its  denominator  is  always  100. 

(2)  This  denominator  may  be  expressed  by  the  sign  %. 
252.    Express  the  following  as  common  fractions,  and  re- 
duce to  lowest  terms. 


1. 

7% 

10. 

83  i% 

19. 

A% 

28. 

n% 

2. 

15% 

11. 

225% 

20. 

12% 

29. 

.ih% 

3. 

21% 

12. 

411% 

21. 

31% 

30. 

.7% 

4. 

36% 

13. 

561% 

22. 

1% 

31. 

.08^% 

5. 

25% 

14. 

831% 

23. 

A% 

32. 

2.25% 

6. 

50% 

15. 

1000% 

24. 

i?% 

33. 

.01% 

7. 

75% 

16. 

465% 

25. 

f% 

34. 

.001% 

8. 

381% 

17. 

116-1% 

26. 

i% 

35. 

.00^% 

9. 

621% 

18. 

1% 

27. 

i% 

36. 

2H%. 

204  NEW  ADVANCED  ARITHMETIC. 

'  RULE. 

To  change  any  per  cent  to  a  common  fraction,  erase  the 
per  cent  sign  and  tvrite  lOO  for  a  denominator, 

253.  Write  each  of  the  expressions  in  Art.  253  as  a  deci- 
mal fraction. 

254.  Express  the  following  common  fractions  with  the  per 
cent  sign,  and  also  as  decimal  fractions: 


3 
100 

6. 

39 

100 

11. 

48 
100 

16. 

1.25 

100 

17 
100 

7. 

12i 
100 

12. 

.02 
100 

17. 

.25 
100 

125 
100 

8. 

100 

13. 

100 

18. 

79 
100 

62^ 

100 

9. 

100 

14. 

75 
100 

19. 

87^ 
100 

250 
100 

10. 

.7 
100 

15. 

.08J 
100 

20. 

.01 
lUO 

255.  There  is  no  problem  in  percentage  that  will  not  fall 
into  one  of  three  general  problems.  A  mastery  of  these 
general  problems  gives  the  technique  of  the  subject. 

256.      FIRST    GENERAL    PROBLEM. 
To  find  any  per  cent  of  any  number. 

RULE. 
Find  one  per  cent  of  the  number  and  multiply  the  result 
by  the  nutnber  of  per  cent. 

257.      ILLUSTRATIVE    PROBLEM. 
Find  18%  of  (521. 

Analysis.  18%  of  a  nnmher  is  -jiffj  of  that  niiinber.  ^-^j  of  G24  ig 
6.24,  wliic'h  is  found  by  making  the  024  stand  two  orders  farther  to  thft 
right,     jij^g  of  624  is  18  times  6.24,  etc. 


PERCENTAGE.  205 

This  method  can  always  be  employed  with  integers  or. 
decimal  fractious. 


Find: 

1.    7%  of  824 

20. 

1?%  of  867 

2.    19%  of  916 

21. 

.2%  of  163 

3.    26%  of  589 

22. 

.25%  of  7826 

4.    35%  of  1230 

23. 

1%  of  7826 

5.   431%  of  1584 

,24. 

.125%  of  5624 

6.    52  J  %  of  6825 

25. 

\%  of  5624 

7.    S&%  of  42563 

26. 

^5%  of  3162 

8.    117%  of  324^ 

27. 

^^%  of  4563 

9.    125%  of  861| 

28. 

.06|%  of  58635 

10.   250%  of  936.8 

29. 

^^%  of  58635 

11.    1000%  of  78.32 

30. 

.061%  of  32064 

12.    17%  of  .4 

31. 

^5%  of  32064 

13.    23%  of  .625 

32. 

.0625%  of  24638 

14.   31%  of  3  J 

33. 

iV%  of  24638 

15.    1%  of  125 

34. 

f  %  of  896.24 

16.    1%  of  324 

35. 

.625%  of  896.24 

17.    1%  of  762 

36. 

T%%  of  756 

18.    ^%%  of  1284 

37. 

41|%  of  756 

19.    hl%  of  825 

38. 

39  i%  of  7824 

258.     Illustrative  Example.     Fine 

7  %  of  |. 

Analysis.    ^^  of  |  =  ^ff^ ;  ^^  of  |  =  ^Vo- 

Find  : 

1.    15%  off.           8. 

76%  of  f|. 

14.    1%  of  ^a. 

2.    16%  off.           9. 

72%  of  21. 

15.    31%  of  ^. 

3.    24  %  of  tV •            N<^™.    21  =  |. 

16.    4^%  of  If. 

4.    30%ofTSj..         10. 

95  %  of  3i. 

.       Note.    \^- X  ^Uxp 

5.    42%  of  if.         11. 

124%  of  15 

17.    81%  offf. 

6.    56  %  of  \^.         12. 

500%ofl5| 

-|.      18.    131%  of  fa. 

7.    63%  off.           13. 

-1%  off. 

19.    182%  of  IH- 

206  NEW   ADVAXCED  ARITHMETIC. 

ORAL    PROBLEMS. 

20.  What  is  A%  of  60?     7%  of  80?     5%  of  90?     12%  of 
400?     11%  of  900? 

21.  What  is  6%  of  25?  of  12?  of  120?  of  1200?  of  15? 
of  150?   of  1500? 

22.  What  is  1%  of  24?  of  36?  of  480?  of  4800?  of  72? 
of  7200? 

23.  What  is  -^^  of  1200?  of  120?  of  12?  off?  of  §? 
off? 

24.  What  is  1%  of  75?  |%  of  640?  ^7o  of  3300?  ^% 
of  1?   1%  off?   1%  of  i§? 

25.  AYhat  is  10%  of  2,500  pounds?  16%  of  $4,000?  7% 
of  71  miles?     §%  cf  120  acres?     J%  of  2,500  bushels? 

259.  Problems  are  often  simplified  by  changing  per  cent 
to  a  common  fraction  in  its  lowest  terms. 

Illustrative  Problem,     l.    Find  37J%  of  96. 

-1  «        374      75  75       3      3     , 

Analysis.     371%= — -  =  —  = =  -.        of  96  =  36. 

^"'        100       2    X  100      200      8      8 

2.  Findl2i%  of  72;  of  144:  of  60  ;  of  240  ;  off;  oi  ^%. 

3.  Find  621%  of  2400;  of  320;  of  .048  ;  of^^^';  of  i»5  ;  of 
84000. 

4.  Find  40%  of  250;  75%  of  f;  87^%  of  ^;  66.2%  of 
.081;  25%  of  16;  6]-  %  of  .32;  8J%  of  132;  50%  of  1;  60% 
of  2. 

5.  Find  37A%  of  96;  of  120;  of  144;  of  4;  of  .64. 

6.  Find  33^%  of  27  ;  of  81 ;  of  122  ;  of  650  ;  of  f  ;  of  .018. 

7.  Find  16§%  of  84 :  of  120;  of  135;  of  225 ;  of  i ;  of 
f^;  of  .0144. 

8.  Find  6§%  of  45;  of  80 ;  of  140;  of  328;  of  ^ ;  of  |f  ; 
of  .18. 

9.  Find  18|%  of  160;  of  324;  31^%  of  256,  of  320; 
433%  of  180,  of  j|. 


PERCENTAGE.  207 

10.  Find  56|%  of  .0288;   68^%  of  /a- 

11.  Find  20%  of  165  ;  of  f  ;  of  .72. 

12.  Find  40%  of  821 ;  80%  of  .096. 

260.     1.    Find  7%  of  325. 

FIRST   FORM. 
325 

.07         For  analysis,  review  Multiplication  of  Decimals. 
22.75 

SECOND    FORM. 

7  1-^        91 

-  X  3?P  ==  ^  -  "'• 

4 

Find : 

2.  9%  of  426.  5.    20%  of  630  bushels. 

3.  13%  of  612.  6.    23%  of  1,824  miles. 

4.  17%ofS725.  7.    33i%  of  756  acres. 

8.  125%  of  67.2  rods. 

9.  37i%  of  t-t;  of  .0688;  of  432. 

10.  2%  of  7563;  %%  of  1200. 

11.  37^%  of  £24  16  s.  8d. 

12.  33.\%  of  15  lb.  9  oz.  18  pwt. 

13.  25%  of  10  rd.  2  ft.  4  in. 

14.  72%  of  75  cwt.  75  lbs. 

15.  75%  of  440  sheep. 

16.  A  cistern  Avith  a  capacity  of  84  barrels  is  41?  %  full. 
How  many  barrels  does  it  contain  ? 

17.  How  much  is  200%  of  a  quantity?    400%?     1000%? 
250%?    75%?    37i%?    83J%?    66^%?    41-|%? 

18.  Find  27%  of  $864.50. 

19.  Find  6?%  of  $965.80. 

20.  What  is  f  %  of  $1,286.43? 

21.  What  is  183%  of  $1,680.48? 


208  NEW  ADVANCED  ARITHMETIC, 

22.  What  is  1%  of  8972.84? 

23.  What  is  21%  of  7,824  bushels? 

24.  A  merchant  bought  a  stock  of  goods  for  $8,324.60. 
The  charge  for  transportation  was  1|%  of  the  cost.  What 
was  the  entire  cost? 

25.  A  owed  B  a  certain  sum  of  money.  After  paying 
him  20%  of  the  debt,  25%  of  the  remainder,  50%  of  what 
then  remained,  and  83  J  %  of  the  third  remainder,  what  part 
of  the  debt  was  still  unpaid? 

26.  What  is  the  interest  on  8468.15  for  one  3'ear  at  7%  ? 

KoTE.  Interest  is  the  amount  paid  for  the  use  of  money,  and  is  com- 
puted at  a  given  per  cent  of  the  amount  loaned,  called  the  principal,  for 
one  year. 

27.  What  is  the  interest  on  81,236.50  fortwoyearsat  6^%  ? 

28.  What  is  the  interest  on  82,580  for  2  years  and  6  months 
at  6| %  ? 

29.  A  farmer  owns  a  section  of  land.  25%  of  it  is 
meadow,  33^%  of  the  remainder  is  corn-land,  37|  %  of  the 
remainder  is  pasture,  80%  of  the  remainder  is  wheat-land, 
and  the  rest  is  oat-land. 

Note.     Make  a  diagram  8  inches  on  a  side,  and  show  the  several  tracts. 

30.  A  piece  of  cloth  containing  36  yards  was  found  to 
have  lost  3^%  of  its  length  by  shrinkage  after  sponging. 
How  much  did  it  lose  in  length? 

31.  A  man's  income  is  Si  ,500.  He  pays  465  %  of  it  for  his 
household  expenses,  20%  of  it  for  general  expenses,  and  13^% 
of  it  for  personal  expenses.  What  are  his  expenses  for  the 
year?     How  much  does  he  save? 

32.  In  a  school  of  650  pupils  52%  were  girls.  How  many 
boys  were  there? 

33.  A  schoolhouse  is  98  feet  long.  Its  width  is  83  J  %  of 
its  length.     How  wide  is  it? 


PERCENTAGE.  209 

34.  In  1895  there  was  a  shrinkage  in  the  value  of  farm 
lands  of  not  less  than  22  %  .  What  reduction  would  this  make 
in  the  value  of  a  farm  of  320  acres  formerly  worth  $85  an 
acre? 

35.  A  merchant  bought  an  overstock  of  goods  which  cost 
him  $12,8G0.  He  marked  them  to  sell  at  an  advance  of  32 
per  cent.  He  finally  sold  them  at  a  discount  of  28%  of  the 
marked  price.     Did  he  gain  or  lose?     How  much? 

36.  The  south  wall  of  the  room  in  which  I  am  writing  is 
15'  X  26'.  It  has  three  rectangular  windows  whose  aggregate 
area  is  30|g%  of  the  wall.  The  windows  are  of  uniform  size, 
the  width  being  40%  of  the  length.  What  is  the  area  of  each 
window?  its  width?  its  length?     (Diagram.) 

261.  The  second  general  problem  of  percentage  is  to  find 
the  per  cent  that  one  number  is  of  another. 

Illustrative  Problem.     8  is  what  per  cent  of  15? 
Two  things  are  to  be  done  in  solving  this  problem. 

1.  We  are  to  find  what  part  8  is  of  15. 

2.  The  resulting  fraction  is  to  be  changed  to  hundredths. 
The  first  will  require  a  review  of  the  method  of  finding  the 

part  that  one  number  is  of  another. 

The  second  will  require  a  review  of  the  methods  of  chang- 
ing a  common  fraction  to  a  decimal. 

Analysis.     8  is  yV  of  15.     -f^  =  .531  =  53^  % . 

Find  what  per  cent  the  first  number  is  of  the  second  in 
each  of  the  following  pairs  : 


1. 

4 

8. 

2. 

5 

20. 

3, 

6 

8. 

4. 

8 

10. 

5. 

11 

20. 

6. 

18 

25. 

7. 

21  : 

30. 

8. 

5 

30. 

9. 

18 

54. 

10. 

42 

63. 

11. 

7 

56. 

12. 

15  . 

24. 

13. 

21 

56. 

14. 

3  : 

36. 

\  is  f 

off. 

1  = 

15. 

5 

35. 

16. 

5 

4. 

17. 

14 

12. 

18. 

24  . 

6. 

19. 

i 

i- 

20. 

1 

^^' 

21. 

\ 

h' 

Note.     1  =  |.    J  =  f .    |  is  f  of  |.    f  =  66f  % .    Hence  i  is  66|%  of  J. 


22. 

f 

:        ; 

23. 

t 

: 

Note. 

.4 

28. 

.018 

210  NEW  ADVANCED  ARITHMETIC. 

24.  Jj  :   |.  26.    i  :  i- 

25.  ^  :   i.  27.    .4  :    .25. 

.40.     .40  is  1^  of  .20.     |4  =  f^g  =  160%. 
.2.  30.    .007i  :'  .03. 

29.    .024  :    .1.  31.    2^  :   3^. 

32.  A  boy  having  20  marbles  lost  3  of  them.  What  per 
cent  of  his  marbles  did  he  have  left? 

33.  In  a  school  of  42  pupils,  7  were  in  one  class,  14  m  a 
second,  6  in  a  third,  12  in  a  fourth,  and  the  remainder  in  a 
fifth.     Give  the  per  cent  of  the  school  in  each  class. 

34.  A  man  owning  |-  of  a  mill  sold  i  of  it.  What  per 
cent  of  his  share  did  he  sell?  What  per  cent  of  the  mill  did 
he  still  own? 

35.  A  received  a  salary  of  $12.5  a  month.  His  board  cost 
him  $20,  his  clothing  $5,  his  other  expenses  $'30.  Each  of 
these  items  is  what  per  cent  of  his  income  ?  He  saves  what 
per  cent? 

262.      WRITTEN    PROBLEMS. 

Illustrative  Problems. 
1.    $17  is  what  per  cent  of  $24? 
Analysis.     $17  is  ^|  of  $24.     \\  is  to  be  reduced  to  hundredths. 

First  Method.  Reduce  the  numerator  to  hundredths,  and 
divide  it  by  the  denominator. 

ix  =  ^_ofl7.     17  =  17.00.     ^:f  of  17.00  = 

24)  17.00  (.70^.     70§  =  70|fc. 

16.8 

.20 

Second   Method.     Do  anything  to  the  fraction   that  will 

make  its  denominator  100  without  changing  the  value  of  the 

/17x4^ 
fraction 


24  X  4J 


=  100  J 


PERCENTAGE. 


211 


70# 
Multiplying  both  terms  by  4^,  |^|  is  found  to  equal  "Tq^; 

hence,  17  is  70-^%  of  24. 

2.  f  is  what  per  cent  of  |  ? 

Analysis.    f=ii.    f  =  fl-    il    is  Jf  of  |f.    Chauge  |f  to  hua. 
dredths. 

3.  2  is  what  per  cent  of  450? 

Analysis.    2  is  :j|^  or  ^\^  of  450.     5^3  =  V|^  =  "^^^  =  .OO*  =  |  %. 
Note.     Abundant  dictation  work  is  needed  to  give  facility. 

Jlethod.     18  is  what  per  cent  of  37? 


37)  18.00  (.4S|4. 
118  ^ 

3.20 
2.96 


24 


Problems  like  this  should  be 
performed  at  the  rate  of  two  or 
three  a  minute.  Keep  the  deci- 
mal point  in  its  proper  place  in 
all  of  the  work. 


263.    EXAMPLES    FOR   PRACTICE. 

Find  what  per  cent  the  first  number  is  of  the  second  tu. 
each  of  the  following  problems : 


1.  18  :  30.  13.  125  :  625. 

2.  14:50.  14=  140:720. 

3.  23:69.  15.   99:451. 

4.  36:81,  16.  328:  1076. 

5.  54  :  88.  17.  256  :  72. 

6.  69:52.  18.  500:  128. 

7.  66:92.  19.  836:  1000. 

8.  72-80.  20.  f:^^. 

Form,  f  x  Y-  =  M  =  28)  45.00 

9.  84:  64.  21.  f  :  i§.        33. 

10.  91:28.  22.  f:H-        34-  24 :  70J. 

11.  96:124.  23.  \:{§.  35.  1% :  .5625. 

12.  29:37.  ?4.  .02:. 25.      36.  3.75:71. 


25.  .15:. 125, 

26.  3.5:7.15. 

27.  2J^:5f. 

28.  7|i  :  24i|. 

29.  .0128:. 456. 
7/^:35f. 
1 


30 
31 
32 


T3- 

3i  :  28. 


212  NEW  ADVANCED  ARITHMETIC. 

37.  $24  :  $84.  47.  2  yd.  2  ft.  3  iu. :  12  rd. 

38.  $63:  $40.  48.  $10.24  :  $1280. 

39.  125  lbs. :  370  lbs.  49.  375  men :  12000  men. 

40.  84  A.:  640  A.  50.  fr^. 

41.  130  sheep:  1200  sheep.     51.  3i :  7|. 

42.  12|  days  :  19  days.  52.  3  qt.  1  pt. :  5  gal.  2  qt. 

43.  lOOG  rd. :  25  rd.  53.  40  sq.  rd. :  8  A. 

44.  6  ft. :  324  ft.  54.  3  yd. :  8  rd. 

45.  136:  624.  55.  2°  30':  10°. 

46.  38:   112.  56.  $624:  $12. 

57.  A  man  bought  a  farm  for  $6,250.  He  paid  cash 
$1,250.  What  per  cent  of  the  purchase  price  remained 
unpaid  ? 

58.  A  man  had  24  cd.  6  cd.  ft.  of  wood.  He  sold  4  cd. 
4  cd.  ft.     AYhat  per  cent  of  his  wood  was  left? 

59.  25%  of  an  article  is  what  per  cent  of  |  of  it? 

60.  40%  of  I  of  an  article  is  what  per  cent  of  all  of  it? 

61.  If  A's  money  is  25%  of  B's  more  than  B's,  B's  is 
■what  per  cent  of  A's  less  than  A's  ? 

Note.  If  A's  is  25%  of  B's  more  than  B's,  it  is  125%  of  B's,  or  f  of 
B's ;  hence,  B's  is  f  of  A's.  (Prove  this.)  If  B's  is  |  of  A's,  it  is  ^  of  A's 
less  than  A's  ;  hence,  is  20%  of  A's  less  than  A's.  Observe  that  "  per  cent 
of  what  1  "  is  the  important  question. 

62.  If  A's  money  is  25%  less  than  B's,  B's  is  what  per 
cent  more  than  A's? 

63.  If  B's  money  is  33)l%  more  than  A's,  A's  is  what  per 
cent  less  than  B's  ? 

Note.     Form  problems  until  the  process  is  mastered. 

64.  If  A's  money  is  10%  more  than  B's,  B's  is  what  per 
cent  less  than  A's?  15%  more?  20%  more?  30%  more? 
40%  more?  50%  more?  37i%  more?  62^%  more?  66^% 
more?     83i%  more? 


PERCENTAGE.  213 

65.  The  width  of  the  top  of  your  desk  is  what  per  cent  of 
its  length  ? 

66.  The  width  of  your  school- room  is  what  per  cent  of 
its  length? 

67.  The  length  of  this  book  is  what  per  cent  of  its  width  ? 

68.  The  thickness  of  this  book  is  what  per  cent  of  its 
width?  of  its  length? 

69.  The  number  of  boys  in  your  room  is  what  per  cent 
of  the  whole  number  of  pupils  in  the  room  ?  It  is  what  per 
cent  of  the  number  of  girls  ? 

70.  The  percentage  of  girls  in  your  room  is  what  per  cent 
of  the  percentage  of  gMs  in  the  primary  room  ? 

71.  What  was  the  percentage  of  boys  in  the  last  gradu- 
ating class  in  your  high  school?  of  girls? 

72.  What  is  the  percentage  of  attendance  in  your  room 
for  one  month  if  all  the  pupils  enrolled  except  five  were 
present  each  day,  and  if  each  of  them  was  absent  three 
days? 

73.  What  is  the  population  of  the  town  or  city  in  which 
you  live?  What  is  the  enrollment  in  public  schools?  AYliat 
per  cent  of  the  population  is  in  school  ? 

264.  The  third  general  problem  of  percentage  is  to  find 
a  number  Twhen  some  per  cent  of  it  is  given. 

Illustrative  Problem.     24  is  6%  of  what  number? 

Analysis.  Since  24  is  6%  of  a  required  number,  1  %  of  that  number 
is  \  of  24,  which  is  4.  100%  of  the  required  number  is  100  times  4,  which 
is  400. 

Method.  Find  1%  of  the  required  number  by  dividing 
the  given  number  by  the  number  of  per  cent.  Multiply  this 
quotient  by  100. 

Such  problems  may  be  taken  out  of  percentage  by  chang- 
ing the  per  cent  to  a  common  fraction;  thus,  6%  =  ^• 
24  is  ^^  of  what? 


214  NEW  ADVANCED  ARITHMETIC. 

265.    ORAL    PROBLEMS. 

1.  48  is  10%  of  what?  12%  of  what?  25%  of  what? 
S3i%  of  what?   50%   of  what? 

2»  60  is  1%  of  what?  |%  of  what?  ^7o  of  what?  ^^%  of 
•what? 

3.  f  is  5%  of  what?  g%  of  what?  25%  of  what?  100% 
of  what?  621%  of  what? 

4.  $21  is  25%  less  than  what? 

5.  $18  is  91%  less  tha-u  what? 

6.  $150  is  50%  more  than  what? 

7.  60  A.  is  30%  of  what? 

8.  75  miles  is  25%  of  what? 

9.  J  of  I  of  a  yard  is  20%  of  what? 

10.    §  of  I  of  ^  of  a  bushel  is  16§%  of  what? 

266.    WRITTEN    PROBLEMS. 

1.  $6.24  is  18%  of  what? 

FORM. 

$6.24  X  100  Employ  eaucellatioa. 

2.  750  rd.  is  125%  of  what? 

3.  18  gal.  3  qt.  1  pt.  is  6|%  of  what? 

4.  $62.50  is  15t%  of  what? 

5.  4  sq.  rd.  16  sq.  yd.  is  5^%  of  what? 
€.  4  cu.  ft.  428  cu.  in.  is  73f  %  of  what? 

7.  67°  30'  is  182%  of  what? 

8.  A  isf%  of  what? 

9.  980  A.  is  51%  less  than  what? 

10,  $6,820  is  241%  more  than  what? 

11.  72  men  deserted  from  a  regiment,  leaving  92f%  ot 
ihe  whole  number.  How  many  men  were  there  in  the  rogi- 
ment  before  the  desertion  ? 


r  ER  CEN  TAGE.  215 

12.  Paid  663.84  for  the  use  of  a  certain  sum  of  money  for 
one  year  at  I'/c     What  was  the  sum  r 

13.  Paid  690  for  the  use  of  a  certain  sum  of  money  for  2 
years  at  the  rate  of  6  %  for  a  year.     What  sum  was  borrowed? 

14.  The  interest  on  a  certain  principal  for  4  years  and  7 
months,  at  Q'/c •>  is  $269.94.     What  is  the  principsd? 

15.  8141,57  is  18%  of  what  number? 

16.  84  rods  is  125%  of  what  number? 

17.  4^^  is  37 i  %  of  what  number? 

18.  5  lbs.  3  oz.  6  pwt.  is  33 i%  of  what? 

19.  35  barrels  is  4I5  %  of  the  capacity  of  a  cistern.  How 
many  barrels  will  it  hold? 

20.  8315.09  is  182%  of  ^'^^^  number  of  dollars? 

21.  The  width  of  a  pane  of  glass  in  m}^  window  is  14 
inches,  which  is  487^^9%  of  the  lengtli.  How  long  are  the 
panes  ? 

22.  AVhat  is  the  area  of  each  pane?  This  is  o\^^^?.2%  of 
what  ? 

23.  Find  the  luinibcr  of  Mhich  263  is  ]  %' ;  of  which  79 
is|%. 

24.  826  is  i\%  of  what  number? 

25.  964  is  500%  of  what  number?  1000%  of  what  number? 
1331%. 

26.  Sold  445.5  pounds  of  sugar,  which  was  44%  of  what 
I  had  left.     How  much  had  I  at  first? 

27.  Sold  a  farm  for  812,860.  f  of  this  amount  is  50%  of 
the  cost  of  the  farm.     The  gain  is  what  per  cent  of  the  cost? 

28.  What  numl)er  increased  by  20%  of  itself  equals  126? 

29.  What  number  diminished  by  20%  of  itself  equals  126? 

30.  A  railway  train,  running  at  an  average  rate  of  35 
miles  an  hour,  for  2f  hours,  passes  over  35%  of  the  conduc- 
tor's run.     ^Miat  is  the  length  of  his  division? 

31.  What  number  increased  by  25%  of  itself  equals  J? 


216  XEW  ADVAXCED  ARITHMETIC. 

32.  AVhat  number  diminished  by  75%  of  itself  equalSj^? 

33.  If  340  be  added  to  a  number,  the  result  is  117%  of  the 
number.     What  is  the  number? 

34.  If  520  be  subtracted  from  a  number,  the  result  is  86% 
of  the  number.     What  is  the  number? 

35.  A  town  is  found  to  have  gained  824:  in  population  in 
5  years.  This  is  an  increase  of  8% .  What  was  the  popula- 
tion of  the  town  5  years  ago  ? 

36.  A  traveler  having  gone  384  miles  has  completed  84% 
of  his  journey.     How  much  farther  has  he  to  go? 

37.  A  farmer  having  plowed  36]  acres  finds  that  he  has 
finished  56%  of  his  field.     How  large  is  it? 

38.  A  farmer  contracted  to  deliver  to  a  dealer  1 ,800  bushels 
of  corn.  Upon  taking  his  grain  to  market  he  found  that  he 
had  overestimated  the  capacity  of  his  crib  4%.  How  much 
did  it  contain  ? 

39.  A  merchant  being  obliged  to  vacate  his  room  sold  his 
stock  at  a  discount  of  11  %  of  the  cost  and  realized  823,568.46. 
What  did  the  goods  cost  him? 

40.  In  a  certain  school  there  are  168  boys,  who  form  42% 
of  the  whole  school.     How  many  girls  are  there  in  the  school  ? 

41.  A  merchant  sold  a  suit  of  clothes  for  S28.50,  which 
was  25%  less  than  the  marked  price.  This  was  33 ;\''^  more 
than  the  cost.     What  was  the  cost? 

42.  $3,825  is  11%  more  than  what?  15%  less  than  what? 

43.  What  number  increased  by  18%  of  itself  equals 
3,379.52? 

44.  AVhat  fraction  diminished  by  30%  of  itself  equals  ^i? 

45.  's  -^  i  equals  24%  of  what? 

4  4-  t'- 

46.  V  e<iuals  14J,%  of  what  number? 


PERCENTAGE.  217 

47.  In  a  certain  city  there  are  15%  more  Germans  than 
Swedes.  The  former  number  5,589,  and  the  latter  form  9% 
of  the  entire  population.     What  is  the  population  of  the  city? 

48.  £18  17  s.  5  d.  is  7%  of  what? 

49.  A  traveler,  having  gone  24  mi.  124  rd.  4  yd.,  has 
completed  37i%  of  his  journey.  What  distance  has  he  yet 
to  travel? 

50.  A  has  a  tract  of  land  containing  82  A.  120  sq.  rd. 
This  is  5%  more  than  B's.     How  much  has  B? 

51.  A  wholesale  merchant's  sales  were  6|%  less  in  1895 
than  in  1894,  when  they  aggregated  $824,960.50.  What 
were  his  sales  in  1895? 

52.  H.  M.  Senseney  lost  by  fire  3,720  tons  of  coal.  This 
was  62%  of  his  stock.     What  was  his  stock? 

53.  H.  M.  Senseney  carried  23%  more  stock  at  the  time  of 
the  fire  than  Parker  Bros.     What  amount  had  Parker  Bros.? 

54.  The  uninjured  portion  of  11.  M.  Senseney's  stock  was 
18%  less  than  the  amount  required  for  a  year's  supply  to  a 
manufacturing  establishment.  How  much  did  it  consume 
annually  ? 

55.  A  traveler  starts  from  Boston  and  goes  west.  When 
he  reaches  Chicago  he  finds  that  he  has  passed  over  31^%  of 
the  longitude  to  be  described  in  his  journey.  His  destination 
is  near  what  city  whose  longitude  is  given  in  the  table? 

56.  63  lb.  12  oz.  avoir,  is  62i%  of  what? 

57.  A  pile  of  wood  6  feet  high,  4  feet  wide,  and  28  feet 
long  is  33^%  of  how  many  cords? 

58.  15  d.  12  hr.  30  min.  is  12%  of  what? 

59.  A  room  18  feet  long,  15  feet  wide,  and  10  feet  high 
contains  75%  of  the  number  of  cubic  feet  of  another  room  of 
the  same  height  and  width.     What  is  its  length? 

60.  $74.88  was  paid  for  the  use  of  a  certain  sum  of  money 
for  two  years  at  the  rate  of  6%  of  the  money  for  its  use  foi* 
one  year.     What  was  the  principal? 


218  XEW  ADVANCED  ARITHMETIC. 

267.       MISCELLANEOUS    PROBLEMS. 

1.  Find  23^%  of  $628.50. 

2.  What  is  the  interest  on  81,296,  for  one  year,  at  5J%  per 
annum  ? 

3.  If  I  paid  §71.28  for  the  use  of  -31,296,  for  one  year, 
what  is  the  rate  of  interest? 

Note.  Rate  is  the  number  of  huudredths  of  tlie  priucipal  j)aid  fur  ltd 
nsft  for  one  year. 

4.  Paid  $225  for  the  use  of  $1,500  for  2  years  and  6 
months.     What  is  the  rate? 

5.  The  lot  upon  which  my  house  is  built  iias  a  frontage  of 
121  feet,  which  is  53^%  of  its  depth.     What  is  its  depth? 

6.  How  many  square  feet  does  the  above  lot  contain? 
"What  part  of  an  acre  is  it?     Change  this  result  to  per  cent. 

7.  The  house  on  the  above  lot  stands  back  OO  feet  from 
the  street.  What  per  cent  of  the  lot  lies  in  front  of  the 
house  ?  " 

8.  My  farm  is  the  northeast  quarter  of  a  section.  How 
many  acres  does  it  contain?  I  |)aid  880  an  acre  for  it.  What 
did  it  cost?  It  has  diminished  in  value  8%.  What  is  its 
present  value  ? 

9.  For  the  use  of  the  above  farm  my  tefiant  pays  me  Jj  of 
the  oats  raised,  ^  of  the  corn,  and  84.75  an  acre  for  meadow 
and  pasture.  Last  year  the  S.  E.  \  was  sowed  in  oats,  the 
west  \  was  planted  in  corn,  and  the  N.  E.  I  was  meadow  and 
pasture.  The  oats  yielded  an  average  of  51  bushels,  and 
sold  for  18  cents.  The  corn  yielded  an  average  of  54  bush- 
•els,  and  sold  for  25  cents.  The  taxes  and  repairs  were  8200. 
What  per  cent  of  its  present  value  did  it  pay? 

10.  17i  is  what  part  of  49?  Change  the  result  to  per 
cent. 

11-    2  is  -nhat  part  of  ^^i  ?     Wiuit  per  cent? 


PER  CENT  A  GE.  21^ 

12.  A  man  earns  $15  a  week.  He  pays  $4.50  for  boards 
70  cents  for  car  fare,  an  average  of  $1.25  for  clothing,  and 
$3.20  for  all  other  expenses.  How  much  can  he  save  in  a 
year?     This  is  equal  to  the  interest  on  what  sum  at  5%  ? 

13.  Multiply  one  thousand  one  ten-thousandths  by  four 
thousand  three  millionths. 

14.  Divide  one  thousand  one  hundred  one  millionths  by 
one  ten-thousandth. 

15.  Add  the  following  innnbers,  and  obtain  the  correct 
result  in  one  minute:"  34883469,  55273289,  52678979, 
46864278,  54489858,  47791697,  34963248,  46815798, 
68866337. 

16  e  Arrange  these  problems  as  given  below,  and  place. 
the  differences  at  the  right,  thus  : 

75063  —  38156  =  36907 
84152  -  68237  = 
91005  -  42807  = 
63254  —  27809  = 
83274  —  58695  = 
91352  -  63806  = 
74083  —  35108  r:^ 

Find  the  sum  of  the  minuends ;  do  the  same  with  the  sub- 
trahends ;  with  the  remainders.  To  the  sum  of  the  subtra- 
hends add  the  sum  of  the  remainders.  If  no  error  has  been 
made,  what  should  the  last  sum  equal? 

17.  A  man  deposited  $8,650  in  a  bank  on  May  1.  During 
the  month  he  drew  the  following  checks  against  his  deposit ; 
$650.70,  $329.85,  $48,  S64..50,  $1,540.90,  $1,937.20,  876.80, 
$2,170.40.     "What  was  his  balance  on  June  1  ? 

18.  The  following  is  a  copy  of  A's  bank  account  for 
June:  Balance  to  his  credit  June  1,  $584.60;  June  2,  de- 
posited $275.25  ;  June  4,  drew  a  check  for  $146.85  ;  June  6, 
deposited  $64.50;  same  day  drew  a  check  for  $186.15;  June 
10,  deposited  $225 ;  June  15,  drew  a  check  for  $324.10,  and 


220  AEW  ADVANCED  ARITHMETIC. 

on  June  18  for  $462.90;  June  21,  deposited  §240;  June  25, 
drew  a  check  for  $72.12.     What  was  his  balance  July  1  ? 

19.  (74  -  16)  -  (62  -  36)  =  ?  (74  -  16)  -  62  -  36  =  ? 

20.  562  +  79  —  (324  +  148)  =  ? 
562  +  79  —  324  +  148  ==  ? 

21.  Bought  of  A  974  bushels  of  oats  at  19^  cents;  1,328 
bushels  of  corn  at  28|  cents;  1,726  bushels  of  wheat  at  58 
cents.  Sold  him  30  acres  of  land  at  039  an  acre.  Which 
was  indebted  to  the  other?     How  much? 

22.  Two  railway  trains  start  at  the  same  time  from  the 
opposite  ends  of  a  division  212  miles  long.  Oue  runs  at  au 
average  rate  of  29  miles  an  hour,  and  the  other  at  24  miles. 
How  far  apart  will  they  be  at  the  end  of  3  hours  ?  4  hours  ? 
5  hours?  6  hours? 

23.  A  stock  train  has  29  cars.  Each  car  coutains  19 
cattle,  whose  average  weight  is  1,450  pounds.  They  sell  for 
$5.25  a  hundred.     What  do  they  bring? 

24.  Change  i\  to  a  5-place  decimal. 

25.  Add  5  mi.  180  rd.  4  yd.  2  ft.;  16  mi.  79  rd.  3  yd. 
1  ft.;  26  mi.  136  rd.  2  yd.  2  ft.  8  in.  ;  29  mi.  278  rd.  5'ft. 
10  in.;  46  mi.  316  rd.  1  yd.  1  ft.  10  in. 

26.  Multiply  2  lb.  10  oz.  16  pwt.  15  gr.  by  36. 

27.  Reduce  10  square  yards  to  a  fraction  of  an  acre. 

28.  If  18  gallons  of  water  be  mixed  with  22^  gallons  of 
grape-juice,  the  water  is  what  per  cent  of  the  mixture  ? 

29.  How  many  barrels  (3H  gallons)  will  a  cylindrical 
cistern  hold,  its  diameter  being  8^  feet  and  its  depth  10 
feet? 

30.  A  delivery  pipe  3  inches  in  diameter  has  what  per- 
centage of  the  capacity  of  a  pipe  whose  diameter  is  3^ 
inches? 

31.  A  man  whose  watch  shows  Chicago  time  finds  that  it 
is  27  min.  36  sec.  slower  than  local  time.  \Vhat  is  his 
longitude  ? 


PERCENTAGE.  221 

32.  How  many  wagon-loads  of  sand,  each  containing 
83^-%   of  a  cubic  yard,  will  fill  your  school-room? 

33.  If  a  pile  of  wood  6  feet  high  and  4  feet  wide  extend 
across  the  front  of  your  school  lot,  what  is  it  worth  at  $4.75 
a  cord? 

34.  What  is  the  altitude  of  a  triangle  whose  area  is  an 
acre,  and  whose  base  is  10  feet? 

35.  A  square  tract  of  land  containing  0,960.6  acres  costs 
$80  an  acre.  The  number  of  silver  dollars  required  to  pay 
for  it  will  exactly  cover  its  boundary:  what  is  the  distance 
around  the  field?     What  is  the  length  of  one  side? 

Note.  "What  is  the  diameter  of  a  silver  dollar?  What  is  the  distance 
around  the  field  in  feet?  What,  then,  is  the  length  of  one  side?  Prove 
the  problem. 

36.  Divide  160  square  rods  into  18  equal  parts. 

37.  What  is  the  cost  of  plank  2  inches  thick  to  build  a 
walk  250  feet  long  and  6  feet  wide  at  $21.50  a  thousand 
feet? 

38.  A's  money  is  25%  less  than  B's  and  25%  more  than 
C's.     If  A  has  $324,  how  much  has  B?     How  much  has  C? 

39.  What  will  discharge  a  debt  of  $586.80  at  a  discount 
of  20%  and  5%  ? 

Note.     The  second  discount  is  computed  on  the  remainder  of  the  first. 

40.  Change  3  inches  to  the  decimal  of  a  rod. 

41.  To  what  single  discount  is  a  discount  of  33.^%  and 
5%  equal? 

42.  1049760  is  the  product  of  three  factors,  two  of  which 
are  216  and  15.     What  is  the  third? 

43.  What  is  the  difference  between  i  %  of  $1,800  and  33. i% 
of  the  same  ? 

44.  A  school-room  is  15'  X  60'  X  72'.  The  ventilator  is 
2'  X  2'.  What  must  be  the  velocity  of  the  air  in  feet  per 
second  to  change  the  air  in  8  minutes? 


222  NEW  ADVANCED  ARITHMETIC. 

45.  The  above  school-room  is  lighted  from  the  long  sides. 
If  the  window  space  is  10%  of  the  floor  space,  how  many 
4'  X  9'  windows  are  there  on  each  side  ? 

46.  The  value  of  a  house  is  87i%  of  the  value  of  the  lot 
Both  cost  $9,645.     What  is  the  value  of  each? 

47.  Put  the  following  in  the  form  of  a  bill,  supplying 
names:  36  collars  at  16|^;  6  shu-ts  at  $1.50;  12  paii's  cuffs 
at  25," ;  15  handkerchiefs  at  30*^' ;  8  pairs  hose  at  45.^  ;  8  ties 
at  35<^ ;  3  suits  underwear  at  S2.50  ;  1  hat  at  S2.50  ;  2  pairs 
gloves  at  $1.25. 

48.  The  width  of  the  blackboard  on  one  side  of  your 
school-room  is  what  per  cent  of  its  length? 

49.  If  a  gallon  measure,  cylindrical  in  form,  is  4  inches  in 
diameter,  what  is  its  height? 

50.  What  is  the  weight  of  a  solid  block  of  gold  l\  feet 
thick  that  will  exactly  cover  the  top  of  vour  teacher's  table? 
(19.4.) 

51.  A  discount  of  20%  and  2^%  having  been  allowed  me, 
my  bill  is  $386.40.     What  was  the  original  bill? 

52.  What  is  the  weight  of  a  dozen  silver  spoons,  each 
weighing  3  oz.  5  pwt.  7  gr.  ? 

53.  Simplify  ^^---^rTTe* 

54.  Write  the  answer  to  the  following  question:  What 
common  fractions  can  be  changed  to  pure  decimals? 

55.  Give  a  rule  for  changing  a  common  fraction  to  per 
cent ;  for  division  of  a  decimal  by  a  decimal ;  for  changing  a 
decimal  to  a  common  fraction  ;  for  finding  the  per  cent  which 
one  number  is  of  another. 

56.  What  is  the  interest  on  $560  for  3  years,  4  months, 

at7%? 


PERCENTAGE.  223 

57.  "What  is  the  cost  of  au  article  which  is  sold  for  $225, 
after  a  deduction  of  10%  from  the  marked  price,  it  having 
been  marked  so  as  to  gain  33^%  ? 

58.  An  article  cost  $7.29.  What  should  be  its  marked 
price  to  permit  a  discount  of  10%  and  still  gain  10%  ? 

59.  How  many  articles,  each  weighing  3  oz.  6  pwt.  20  gr., 
can  be  made  from  9  lb.  8  oz.  19  pwt.  4  gr.  of  the  same 
material  ? 

60.  Find  the  time  from  March  12,  1891,  to  Jan.  5,  1896. 


224  NEW  ADVANCED  ARITHMETIC. 

SECTION  yiii. 

268.     APPLICATIONS    OF    PERCENTAGE. 

1.  The  methods  of  ealcuhition  taught  in  percentage  are 
applied  to  the  solution  of  practical  problems.  In  all  such 
problems  the  pupil  must  determine  which  of  the  three  general 
problems  is  involved. 

2.  In  each  of  the  general  problems  three  numbers  are  em- 
ployed. They  are  called  the  Base,  the  Percentage,  and  the 
Rate  Per  Cent. 

3.  The  Base,  in  a  problem  in  percentage,  is  the  number  to 
which  the  other  two  numbers  are  referred.  It  is  the  answer 
to  the  question,  "  per  cent  of  what?  " 

4.  The  Rate  per  Cent  is  the  decimal  fraction  expressed  in 
hundredths  which  shows  the  part  which  the  percentage  is  of 
the  base. 

5.  The  Percentage  is  the  number  whose  relation  to  the 
base  is  expressed  by  the  rate  per  cent. 

6.  In  each  of  the  general  problems  two  of  these  three 
numbers  are  given  to  find  the  third. 

7.  In  the  first  general  problem  the  base  and  rate  per  cent 
are  given  to  find  the  percentage ;  in  the  second,  the  base  and 
percentage  to  find  the  rate  per  cent ;  in  the  third,  the  per- 
centage and  rate  per  cent  to  find  the  base. 

Note.  A  fourth  number,  called  the  Amount,  is  employed  in  some  of 
the  Applications  of  Percentage. 

269.     PROFIT   AND   LOSS. 

1.  The  Cost  of  an  article  is  the  expenditure  involved  in 
its  purchase  or  production,  and  is  usually  expressed  in 
money. 


APPLICATIONS  OF  PERCENTAGE.  225 

2.  The  Selling  Price  of  an  article  is  the  amount  of  money 
which  the  bu\-er  pays  the  seller  for  it. 

3.  The  Profit  on  an  article  is  the  excess  of  the  selling  price 
over  the  cost. 

4.  The  Loss  on  an  article  is  the  excess  of  the  cost  over 
the  selling  price. 

270.  In  each  ot  the  following  problems  show  which  of  the 
general  problems  is  involved.  Be  ready  to  tell  "per  cent 
of  what"  in  every  case. 

ORAL    PROBLEfylS. 

1.  Paid  $2.50  for  an  article,  and  sold  it  at  an  advance  of 
10%.     "What  was  my  gain? 

2.  Bought  an  article  for  82,  and  sold  it  for  SI. 75.  The 
loss  was  what  per  cent  of  the  cost? 

3.  Gained  30  cents  on  an  article,  which  was  15%  of  the 
cost.     What  was  the  cost? 

4.  Sold  an  article  for  §12,  gaining  20%  of  the  cost.  What 
was  the  cost? 

271.  These  four  problems  illustrate  all  of  the  cases  that 
arise  in  Profit  and  Loss.     They  are  as  follows : 

1.  Given  the  cost  and  the  rate  per  cent  of  profit  or  loss, 
to  find  the  profit  or  loss,  or  selling  price. 

2.  Given  the  cost  and  the  profit  or  loss,  to  find  the  rate 
per  cent  of  profit  or  loss. 

3.  (a)  Given  the  profit  or  loss  and  the  rate  per  cent  of 
profit  or  loss,  to  find  the  cost,  or 

(h)  Given  the  selling  price  and  rate  per  cent  of  profit  ot 
loss,  to  find  the  cost. 


226  N^EW  ADVANCED  APdTHMETIC. 

272.      ORAL   PROBLEMS. 

1.  Cost,  $12;    rate  of  gain,  25^.  :    find  gain  and  selling 
price. 

2.  Cost,  625 ;  gain,  88  :  find  rate  per  cent  of  gain  and 
selling  price. 

3.  Rate  of  loss,  40 /r  ;    loss,  820:    find  cost  and  selling 
price. 

4.  Rate  of  gain,  12i%;  selling  price,  8-4o:  find  cost  and 
gain. 

5.  Cost,  81.44;  loss,  18  cents  :  find  rate  per  cent  of  loss 
and  selling  price. 

6.  Rate  of  gain,  \%l%  \  selling  price,  863:  find  cost  and 
gain. 

7.  Cost,   8^6;    rate  of  loss,   6'^^  :    find  loss  and  selling 
price. 

8.  Cost,  863  ;  gain,  87  :  find  rate  per  cent  of  gain  and 
selling  price. 

9.  Cost,  8540  ;  selling  price,  8600  :  find  gain  and  rate  per 
cent  of  gain. 

10.  Cost,  8225;  rate  of  loss,  11^  7f  :  find  loss  and  selling 
price. 

11.  Gain,  891;    rate  of  gain,   1%  :   find  cost  and  selling 
price. 

12.  Cost,  8250;  rate  of  gain,  8%  :  find  gain  and  selling 
price. 

13.  Cost,  836 ;  selling  price,  842  :  find  gain  and  rate  per 
cent  of  gain. 

14.  Selling  price.  852 ;  cost,  848  :  find  gain  and  rate  p*^r 
cent  of  gain. 

15.  Selling  price,  8560;  rate  of  loss,  20%  :  find  cost  and 
loss. 

16.  Rate  of  gain,  30%  ;  selling  price,  8390:  find  cost  and 
gain. 


APPLICATIONS   OF  PERCENTAGE.  227 

17.  Gain,  $81;  rate  of  gam,  lljj%:  find  cost  and  selling 
price. 

18.  Loss,  $21 ;  cost,  $63  :  find  rate  per  cent  of  loss  and 
selling  price. 

19.  Cost,  $1,800  ;  rate  of  gain,  Q&%%  :  find  gain  and  sell- 
ing price. 

20.  Selling  price,  $3,000;  cost,  $1,800:  find  gain  and 
rate  per  cent  of  gain. 

21.  Loss,  $250 ;  selling  price,  $750 :  find  cost  and  rate 
per  cent  of  loss. 

22.  Gain,  46  cents;  rate  of  gain,  23%  :  find  cost  and 
selling  price. 

23.  Cost,  84  cents-,  rate  of  gain,  1\%  :  find  gain  and 
selling  price. 

24.  Gain,  6  cents;  rate  of  gain,  1\%  :  find  cost  and 
selling  price. 

273.      WRITTEN    PROBLEMS. 

1.  A  man  invests  $2,680,  and  makes  a  profit  of  23|%. 
How  manjf  dollars  did  he  gain  ? 

2.  A  farmer  raised  in  a  single  year  6,400  bushels  of  oats, 
6,250  bushels  of  corn,  and  42  tons  of  hay.  He  sold  the  oats 
at  37i  cents  a  bushel,  the  corn  at  40  cents,  and  the  hay  at 
$6  a  ton.  The  landlord  received  f  of  the  value  of  the  crop, 
which  was  7%  of  the  value  of  the  farm.  What  was  the  farm 
worth  ? 

3.  A  house,  costing  $12,250,  was  destroyed  by  fire.  The 
insurance  company  paid  the  owner  $10,500.  He  lost  what 
per  cent  of  his  investment? 

4.  A  merchant  marked  his  goods  an  advance  of  25  %  above 
the  cost.  The  market  declining,  he  reduced  the  selling  price 
10%,  and  made  a  profit  of  $1,280  on  his  sales.  That  was 
the  cost  of  the  goods  sold? 


228  NEW  ADVANCED  ARITHMETIC. 

5.  If  I'j  of  an  article  be  sold  for  what  the  whole  cost, 
what  is  the  rate  per  cent  of  gain  ? 

6.  A  man  bought  a  quantity  of  apples  at  75  cents  a 
bushel.  If  there  was  a  waste  of  12%,  at  what  price  must 
he  sell  the  remainder  to  make  25%  in  the  transaction? 

7.  A  merchant  sold  450  yards  of  cloth  at  a  profit  of  24%, 
realizing  a  profit  of  $243.  Find  the  cost  and  selling  price 
per  yard. 

8.  Find  the  selling  prices  of  the  following  articles  so  that 
they  may  bring  a  profit  of  30%  :  tea,  costing  50  cents; 
coffee,  costing  30  cents;  shoes,  costing  $2;  cloth,  costing 
$1.50;  eggs,  costing  15  cents;  butter,  costing  20  cents. 

9.  If  a  span  of  horses,  costing  $465,  be  sold  for  $620, 
what  is  the  rate  per  cent  of  gain  ? 

10.  If  the  purchaser  sold  the  same  horses  for  $520.80, 
what  was  the  rate  per  cent  of  loss? 

11.  A  field,  containing  125  acres,  yielded  6,562^  bushels 
of  corn.  The  succeeding  year  the  crop  diminished  4^6  %. 
What  was  the  yield  per  acre  the  second  year? 

Note.    Explain  the  following  method  of  solution : 
65625  bu.  X  2000  _  , 
10  X  125  X  2100 - 

12.  A  merchant  paid  $1,458  for  some  goods.  What  must 
he  ask  for  them  that  he  may  fall  10%,  lose  10%  of  the  sell- 
ing price,  and  gain  10%  in  the  transaction? 

Note.     Tell  "  per  cent  of  what "  in  each  case. 

^     ,.,,..           $1458X11  X  10X10 
Explain  this  form :  ^p  X  9  X  9 =  " 

13.  Sold  5  of  an  article  for  |  of  the  cost  of  the  whole. 
What  was  the  rate  per  cent  of  gain? 

14.  Sold  f  of  an  article  for  ^  of  the  cost  of  the  whole. 
Was  there  a  gain  or  a  loss  ?     Find  the  rate  per  cent. 

15.  Cost,  $8,625 ;  gain,  $2,125  :  find  rate  per  cent  of  gain. 


APPLICATIONS   OF  PERCENTAGE.  229 

16.  Selling  price,  S9,850 ;  cost,  $10,000  :  fiud  rate  per 
cent  of  loss. 

17.  50  acres  of  oats  yielded  an  average  of  45  bushels  last 
year.  This  year  the  aggregate  yield  from  the  same  field  is 
2,500  bushels.     AVhat  is  the  per  cent  of  gain? 

18.  A  merchant  bought  350  bushels  of  potatoes  at  45 
cents.  He  lost  15%  of  them.  For  how  much  a  bushel  must 
he  sell  the  remainder  to  gain  25%  of  the  cost  of  the  whole? 

19.  Bought  a  bankrupt  stock  for  $3,280.  Sold  it  for 
$4,100,  and  discounted  the  bill  5%  for  cash.  What  was  the 
gain  per  cent? 

20.  In  our  school  there  was  a  total  enrollment  last  year 
of  560.  This  year  it  amounted  to  610.  What  is  the  per 
cent  of  gain? 

21.  Cost,  S28.60;  rate  of  gain,  23%.  Find  selling  price 
without  finding  the  gain. 

22.  Cost,  8824.60;  gain,  $206.15.  Find  per  cent  of  gain 
and  selling  price. 

23.  Gain,  $2,400;  rate  percent  of  gain,  16.  Find  selling 
price  without  finding  cost. 

24.  Selling  price,  8860;  rate  of  profit,  25%.  Find  the 
profit  without  finding  the  cost. 

25.  Selling  price,  8428.50;  rate  of  loss,  15%.  Find  the 
cost. 

26.  A  farmer  raised  6,824  bushels  of  grain.  By  fertiliza- 
tion he  increased  the  yield  8J%,  What  was  the  increased 
yield? 

27.  Owing  to  a  storm,  a  railway  train  was  obliged  to  re- 
duce its  average  speed  from  48  to  40  miles  an  hour.  What 
was  the  per  cent  of  reduction?  The  ordinary  time  for  run- 
ning over  the  division  was  4  hrs.  24  min.  How  far  behind 
time  would  the  train  be  upon  reaching  its  destination? 

IRA 


230  NEW  ADVANCED  ARITHMETIC. 

28.  The  gross  gain  of  A's  business  for  one  year  was 
$12,860,  which  is  25%  of  his  investment.  He  paid  for  rent 
$2,000;  for  insurance  $100;  for  clerk  hu-e  $4,000.  Count- 
ing his  own  services  at  $2,000,  what  per  cent  of  his  invest- 
ment did  he  gain  ? 

29.  Sold  a  tract  of  land  for  $8,403.90  which  cost  $9,810. 
"What  was  tlie  per  cent  of  loss? 

30.  Invested  the  proceeds  of  the  above  sale  in  corn, 
which  was  sold  at  an  advance  of  20%.  Was  there  a  gain  or 
a  loss  in  the  whole  transaction?     What  per  cent? 

Note.     In  the  following  problems  obtain  per  cent  approximately. 


< 

Z!ost  Price. 

Selling  Price. 

Gain. 

Loss. 

Per  Cent 

31. 

$824 

$1050 

? 

? 

32. 

V 

$1068 

$124 

? 

33. 

$1580 

? 

$312.50 

? 

34. 

9 

? 

$21.60 

15 

35. 

? 

$2175 

$240 

? 

36. 

$3812 

$3500 

? 

? 

37. 

? 

? 

$14.50 

H 

38. 

? 

$4840 

$650 

? 

39. 

V 

V 

$88.60 

12^ 

40. 

$6000 

7 

? 

H 

41. 

p 

$79.60 

? 

25 

42. 

? 

? 

$29.80 

48 

43. 

$45.20 

$51 

? 

? 

44. 

? 

$4850.80 

$212.20 

V 

45. 

? 

? 

$68.40 

14? 

46. 

$3874.80 

7 

? 

37^ 

47. 

? 

$5160 

$1180 

? 

48. 

? 

y 

$225 

65     i 

APPLICATIONS   OF  PERCENTAGE.  231 


274.    COMMISSION. 

1.  Commission  is  compensation  paid  by  a  person  or  per- 
sons to  a  person  or  persons  for  performing  certain  business 
transactions. 

2.  The  party  for  whom  the  business  is  transacted  is 
called  the  principal. 

3.  The  party  performing  the  service  for  the  principal  is 
called  an  agent,  or  commission  merchant. 

4.  The  services  performed  by  agents  or  commission  mer- 
chants are  of  two  kinds  : 

a.  Those  in  which  money  comes  into  their  hands  to  be 
remitted  to  their  principals. 

6.    Those  in  w-hich  money  is  expended  for  their  principals. 

5.  Agents  may  receive  money  for  their  principals  by 
collecting  debts  due  them,  or  by  selling  property  for  them 
and  receiving  the  proceeds.  They  may  expend  money  for 
them  by  making  purchases  for  them,  or  by  paying  their 
obligations. 

6.  Commission  is  one  of  the  applications  of  percentage 
because  the  agent's  compensation  is  some  per  cent  of  the 
amount  collected  or  expended. 

7.  As  the  agent  renders  two  kinds  of  service,  there  will 
be  two  kinds  of  problems  in  Commission:  the  "selling  or 
collecting"  problems,  and  the  "  purchasing  or  paying"  prob- 
lems. The  former  are  simple  and  easily  solved ;  the  latter 
are  more  difficult. 

8.  The  Commission  is  the  amount  which  the  agent  re- 
ceives for  his  services.  The  Proceeds  are  the  difference 
between  the  Base  and  Commission. 

9.  The  important  thing  to  remember :  The  Base  in  com- 
mission problems  is  the  amount  actually  collected  or  ex- 
pended for  the  principal. 


232  NEW  ADVANCED  ARITHMETIC. 

275.      PROBLEMS. 

1.  A  commission  merchant  sold  12,560  bushels  of  corn  for 
me  at  36  cents.  His  commission  was  2%.  What  was  his 
commission,  and  what  amount  should  he  remit? 

Which  of  the  general  problems  of  percentage  is  this? 

2.  A  man  sent  to  his  agent  $1,050  with  which  to  purchase 
books.  After  deducting  his  commission  at  o%,  how  much 
did  he  expend  in  books?     What  was  his  commission? 

Note.  Uo  not  make  the  mistake  of  calculating  the  commission  on 
$1,050.     This  is  not  the  amount  to  be  expended  for  books. 

Analysis.  The  amount  to  be  expended  for  books  is  100%  of  itseK. 
The  commission  is  5%  of  that  amount.  Their  sum  is  105%  of  the  amount 
to  be  invested  in  the  books.  But  their  sum  is  $1,050;  hence,  $1,050  is  105% 
of  the  amount  to  be  tlius  invested.  1  %  of  the  investment  is  yj;  of  $1,050, 
which  is  $10.  100%  of  tiie  investment  is  100  X  $10,  which  is  $1,000. 
.$1,050  —  $1,000  —  tne  commission. 


!jll0.50X  100  -^,,,  ^,^^ 

■ r^ =  $1000  ;  or,  o^r =  $1000 

10a  " 

^VTiich  of  the  general  problems  is  this  ? 

Note.  It  is  desirable,  often,  to  find  the  commission  alone.  In  the 
above  problem  the  amount  sent  is  \U  of  the  amount  of  the  purchase.  The 
commission  is  t^^^  of  the  amount  of  the  purchase.  The  commission,  there- 
fore, is  y§3,  or  2x  of  the  amount  sent. 

3.  Find  the  commission  on  a  sale  of  $2,150  at  2^% . 

4.  On  $3,184.36  at  3,\%.  6.    On  $236,124  at  .5%. 

5.  On  $46,912.60  at  1|%.         7.    On  $875,635  at  ^\%. 

8.  Amount  sent,  $2,530  ;  rate  of  commission,  2i%  .  Find 
amount  paid  out. 

9.  Amount  sent,  $3,642  ;  rate  of  commission,  3J%.  Find 
amount  invested. 

10.  Amount  sent,  $8,156;  rate  of  commission,  4%.  Find 
amount  invested. 


APPLICATIONS   OF  PERCENT  AGE.  233 

11  Amount  sent,  S2,4;82.36;  rate  of  commission,  2%. 
Find  amount  invested. 

12.  In  Problems  3,  4,  5,  6,  find  the  proceeds  of  the  sales 
without  finding  the  commission. 

13.  In  Problems  8,  9,  10,  find  the  commission  Tvithout 
finding  the  amount  invested. 

14.  An  agent's  commission  for  selling  a  house  and  lot  was 
S240.  The  proceeds  of  the  sale  were  87,760.  What  was 
the  rate  per  cent  of  the  commission  ? 

15.  An  agent's  commission  on  a  sale  was  $324.60;  the 
rate  of  commission  was  3%.     "What  was  the  amount  of  the 

sale  ? 

16.  A  sent  81,250  to  an  agent  with  which  to  buy  a  town 
lot  and  pay  his  commission  at  2^%.  "What  was  the  cost  of 
the  lot?     "What  was  his  commission? 

17.  An  agent  received  a  consignment  of  corn  amounting 
to  2,864  bushels.  He  sold  it  at  39  cents  a  busheL  He  paid 
freight,  875.92,  and  charged  2^';^  commission.  "What  was  his 
commission,  and  what  amount  did  he  remit  to  the  principal? 

18.  An  agent  sold  3,150  bushels  of  wheat  at  92  cents. 
The  net  proceeds  of  the  sale  were  82,836.41|.  What  wag 
the  rate  per  cent  of  his  commission? 

19.  Sold  cotton  on  a  commission  of  4%.  The  commission 
amounted  to  8384.48.  Invested  the  net  proceeds  in  wheat, 
less  a  commission  of  3%.  "Wl:at  was  the  amount  of  the 
second  commission? 

20.  Sold  the  X.  i  of  the  S.  E.  i,  and  the  S.  ^  of  the  N.  E. 
I  of  a  section  of  land,  at  S72jV  an  acre.  My  commission  was 
3^%.  "What  was  the  amount  of  my  commission?  Find  the 
net  proceeds. 

21.  A  principal  sent  to  his  ag^^nt  324  barrels  of  flour,  with 
directions  to  invest  the  proceeds  of  the  sale  in  wheat,  after 
deducting  his  commission  of  2i%   for  selling;,  and  2i%  for 


234  NEW  ADVANCED  ARITHMETIC. 

buying.     The  flour  sold  for  $6.25  per  barrel.     How  man^ 
bushels  of  wheat  at  83^  cents  could  the  agent  buy? 

22.  What  is  an  agent's  commission  for  collecting  Si  ,2G4odO 
at  1|%  ?     What  amount  should  be  sent  to  the  principal? 

23.  An  agent  is  to  pay  a  debt  of  S836  for  a  merchant. 
What  amount  should  the  merchant  remit,  if  the  agent's  com- 
mission is  ^%  ? 

24.  An  agent's  commission  on  an  investment  at  If  %  was 
$52.  What  amount  should  be  sent  him  to  cover  investment 
and  commission? 

25.  What  is  the  commission  on  a  sale  of  S954.80  at  3%  ? 
If  the  proceeds  include  an  investment  and  the  commission  on 
it  at  2%,  what  is  the  commission?  If  the  purchase  were  sold 
at  an  advance  of  20%,  and  the  agent's  commission  for  selling 
were  3  % ,  what  amount  should  be  remitted  to  the  principal  ? 

276.      COMMERCIAL    DISCOUNT. 

1.  Manufacturers  and  wholesale  dealers  usually  issue 
schedules  of  the  prices  of  their  goods.  These  schedules  are 
called  price  lists. 

2.  These  price  lists  do  not  usually  give  the  selling  price, 
but  give  bases  upon  which  discounts  are  made. 

3.  The  deduction  from  the  list  price  of  goods  is  called 
Commercial  Discount,     It  is  usually  computed  in  per  cent. 

4.  As  Commercial  Discounts  are  made  for  different  rea- 
sons, as  the  amount  purchased,  the  time  of  payment,  more 
than  one  discount  is  sometimes  allowed. 

5.  When  more  than  one  discount  is  allowed,  the  first  is 
computed  upon  the  list  price,  the  second  upon  the  remainder, 
and  so  on. 

6.  The  amount  of  a  bill  after  all  discounts  are  withdrawn 
is  called  the  net  amount. 


APPLICATIONS   OF  PERCENTAGE.  235 

277.      PROBLEMS. 

X,  Bought  a  piano  listed  at  $950.  Discount  45%,  and  5% 
for  cash.     AVhat  was  the  net  amount? 

2.  If  the  list  price  had  been  $880,  and  the  discounts  48% 
and  6  % ,  what  would  the  net  amount  have  been  ? 

3o  Rented  a  piano,  listed  at  $1,080,  at  $6  a  month,  with  the 
agreement  that  the  rent  should  be  applied  to  the  purchase 
price  if  I  decided  to  buy  it.  Kept  it  8  mouths,  and  bought 
it  at  a  discount  of  40,  8,  and  4io  What  did  I  pay  in  addition 
to  the  rent  ? 

4.  Find  the  net  amount  of  a  bill  of  $1,524.60,  the  dis- 
counts being  16  and  4. 

5.  Find  the  net  amount  of  a  bill  of  $825,  the  discounts 
being  10,  4,  and  2. 

6.  Which  would  you  prefer,  a  single  discount  of  20 7e,  or 
a  discount  of  I65  and  4? 

7.  Which  is  greater,  a  single  discount  of  25%,  or  a  dis- 
count of  20  and  6]  ? 

80  Find  the  net  amount  of  a  bill  of  $650,  discounted  at 
18|%. 

9.  What  must  be  the  second  discount  on  the  above  bill  ^o 
give  the  same  net  amount  if  the  first  discount  were  12i%  ? 

10.  Find  the  net  amount  of  a  bill  of  $2,500,  discounted  at 
20,  10,  and  5. 

11.  If  the  first  and  second  discounts  in  the  above  bill 
were  25  and  65,  what  must  the  third  be  to  give  the  same  net 
amount? 

12.  The  net  amount  of  a  bill  discounted  at  15  and  10  is 
$722.16.     What  is  the  gross  amount  of  the  bill? 

13.  Find  net  amount  of  a  bill  for  $1,260.50  discounted 
at  81%. 

14.  Wliat  is  the  bill  which,  discounted  at  5|%,  yields  a 
net  amount  of  $648.09? 


236  NEW  ADVANCED  ARITHMETIC. 

15.  The  bill  is  8728.50;  the  net  amount,  $684.79.  What 
is  the  rate  of  discount  ? 

16.  The  amount  of  the  bill  is  6800 ;  the  net  amount, 
$G08 ;  the  first  discount,  20% .     What  is  the  second  discount? 

Find  the  net  amount  of  the  following  bills : 
1.7.    8325.56  at  18  and  3. 

18.  8464.92  at  21  and  6. 

19.  8681.20  at  15,  5,  and  3. 
iO.    8760.10  at  20,  8,  and  2. 

21.  Si, 241. 10  at  12,  8,  and  4. 

22.  200  yards  of  calico  at  6  cents;  225  yards  of  muslin 
at  7  cents;  50  j'ards  of  broadcloth  at  83.50 ;  80  yards  of  silk 
at  81.40;  95  yards  of  flannel  at  39  cents;  72  yards  of  cash- 
mere at  98  cents;  140  yards  of  gingham  at  9  cents.  The 
discounts  were  16  and  5. 

23.  30  sacks  of  flour  at  86  cents;  120  pounds  of  coffee  at 
28  cents ;  60  pounds  of  butter  at  24  cents ;  50  pounds  of 
lard  at  7  cents;  36  pounds  of  tea  at  54  cents;  300  pounds 
of  sugar  at  6  cents;  75  pounds  of  cheese  at  11  cents.  Dis- 
counts, 20  and  3. 

24.  864  bushels  of  corn  at  26i  cents;  1,040  bushels  of 
oats  at  18  cents;  1,500  bushels  of  wheat  at  63  cents;  980 
bushels  of  rye  at  46  cents;  836  bushels  of  potatoes  at  38 
cents:  17  tons  of  timothy  at  86.75.     Discounts,  18^  and  5. 

25.  To  what  single  discount  is  a  discount  of  20  and  5 
equal?     25  and  10?     16|  and  20?     25,  10,  and  4? 

278.     STOCKS,    BONDS.    BROKERAGE. 

1.  Commercial  enterprises  are  often  undertaken  bj  asso- 
ciations consisting  of  a  number  of  individuals. 

2.  If  these  individuals  enter  into  an  agreement  by  con- 
tract, the  association  is  called  a  Partnership,  and  each 
individual    is  called   a  Partner.      If  they  organize  bv  the 


APPLICATIOXS    OF  PERCENTAGE.  237 

election  of  such  otlicers  as  a  president,  secretary,  treasurer, 
and  board  of  directors,  and  secure  a  charter,  the  association 
is  called  a  Corporation,  or  Stock  Company. 

3.  The  Charter  of  a  corporation  determines  its  name,  its 
object,  the  number  of  shares  that  shall  comprise  its  capital 
stock,  the  method  of  managing  its  business,  etc. 

4.  Charters  are  laws  passed  by  legislative  bodies,  or  are 
issued  in  accordance  with  law  by  some  state  officer. 

5.  Railroad,  steamboat,  or  manufacturing  corporations 
are  illustrations  of  stock  companies. 

6.  The  shares  are  usually  one  hundred  dollars  each,  and 
the  document  which  certifies  that  a  person  owns  one  or  more 
of  these  shares  is  a  Stock  Certificate.  Such  certificates  are 
often  called  Stock,  or  Stocks. 

7.  The  aggregate  amount  of  the  shares  of  a  corporation  is 
its  capital  stock. 

8-  The  market  value  of  stock  is  the  amount  for  which  it 
will  sell.  It  is  usually  indicated  by  the  number  of  dollars 
for  which  a  single  100-dollar  share  will  sell.  Thus,  when 
stock  is  quoted  at  108,  the  price  is  S108  for  a  100-dolIar 
share. 

9.  The  par  value  of  stock  is  the  number  of  dollars  for 
which  the  certificate  calls. 

10.  When  the  market  value  exceeds  the  par  value,  the 
stock  is  at  a  Premium.  When  the  two  values  are  equal,  the 
stock  is  at  Par.  When  the  market  value  is  less  than  the  par 
value,  the  stock  is  at  a  Discount. 

11.  AYhen  a  corporation  is  prosperous,  its  income  exceeds 
its  expenses;   it  then  has  a  Net  Income. 

12.  A  Dividend  is  a  division  of  the  net  income  among  the 
stockholders. 

13.  An  Assessment  is  a  sum  levied  upon  the  stockholders 
to  meet  expenses  not  otherwise  provided  for. 


238  NEW  ADVANCED  APdTHMETIC. 

14.  Dividends  and  assessments  are  always  computed  at 
some  per  cent  of  tlie  par  value. 

15.  Stock  certificates  yield  an  income  to  their  owners  only 
when  dividends  are  declared. 

16.  Bonds  are  notes  issued  by  corporations  and  secured 
by  mortgages  on  its  capital  stock.  They  bear  a  specified 
rate  of  interest. 

17.  The  most  common  forms  of  bonds  are  goverrmient 
bonds,  railroad  or  city  bonds,  etc. 

18.  A  Broker  is  a  person  whose  business  is  the  purchase 
and  sale  of  stocks  and  bonds. 

19.  The  broker's  commission  is  called  Brokerage,  and  is 
recTioned  upon  the  jxir  value  of  the  stocks  or  bu7ids. 

How  does  it  differ  from  the  compensation  of  the  commis- 
sion merchant? 

20.  AVhat  is  the  discount  or  premium  when  stocks  or 
bonds  are  sold  at  94?  62?  108?  431?  2171?  65i?  ISGfr 
111?   329^? 

21.  What  is  the  market  price  when  stocks  or  bonds  are 
sold  at  7%  discount?  16^%  premium?  100%  premium? 
43^%  discount?  260%  premium?  1000%  premium?  1% 
discount?     16|%   discount? 

22.  Be  ready  to  recognize  each  of  the  general  problems  of 
percentage  in  the  following  problems. 

279.     PROBLEMS. 

1.  What  is  the  cost  of  25  railroad  shares  at  92,  brokerage 

li%? 

„            „             SlOO  X  2.-)  X1S7 
Short  Form.    :^— j • 

2.  What  is  the  income  from  the  above  stock,  if  it  yields 
an  aunual  dividend  of  4^%  ? 

^           -^  "         SlOO  X  2.5  X  9 
Short  Iorm.     „  .c . 


APPLICATIONS  OF  PERCENTAGE.  239 

3.  A  broker  bought  for  his  principal  16  United  States 
500-dolLar  bonds,  bearing  4%  interest,  at  102. 

(a)    If  the  brokerage  was  2%,  what  was  the  entire  cost? 
(6)    "What  was  the  income  from  the  bonds  ? 
(c)    The  first  year's   income  was  what  per   cent   of   the 
entire  cost?     Show  short  forms  for  a  and  b. 

4.  A  bought  40  shares  of  bank  stock  at  140.  If  the  bank 
declared  semi-annual  dividends  of  4%,  what  was  the  annual 
income  from  the  stock  ?  What  rate  per  cent  of  his  invest- 
ment did  he  receive  annually  ? 

5.  A  invests  $36,000  in  railroad  stock  at  90.  It  yields 
2%  semi-annual  dividends.  What  is  his  annual  income  from 
the  stock? 

Shout  Form.    ?!^^00x100x_4 
90  X  100 

6.  Find  the  brokerage  on  the  following  purchases : 
28  shares  bank  stock  at  125,  brokerage  lf%. 

148  shares  railroad  stock  at  76,  brokerage  2^%. 
86  city  bonds,  brokerage  3%. 
Note.    Brokerage  is  reckoned  on  what  ? 

7.  What  is  the  rate  per  cent  of  income  from  4%  bonds 
bought  at  80?  From  6%  bonds  bought  at  120?  From  5% 
bonds  bought  at  105?     From  4^%  bonds  bought  at  90? 

Find  the  cost  of  the  following  shares  : 

8.  300  Atchison,  Topeka,  and  Santa  Fe  R.  R.  at  14, 
brokerage  \%. 

9.  200  Chesapeake  &  Ohio  at  14|,  brokerage  ^%. 
Q  10.    8,075  Chicago  Gas  Co.  at  57|,  brokerage  tV^- 

11.  16,340  Chicago,  Milwaukee,  &  St.  Paul  R.  R.  at  75^, 
brokerage  tV^* 

12.  5,948  Chicago,  Rock  Island,  &  Pacific  R.  R.  at  63|, 
brokerage  \%. 

13.  276  Lake  Erie  «fe  Western  R.  R.,  preferred,  at  70|, 
brokerage  4%. 


240  liEW  ADVANCED  ARITHMETIC. 

14.  1,300  Illinois  Central  R.  R.  at  93,  brokerage  |%, 

15.  600  Philadelphia  &  Reading  R.  R.  at  13  g,  brokerage 

16.  9,164:  Chicago,  Burlington,  &  Quiucy  at  72|,  broker- 
age ^%. 

17.  1,154;  Western  Union  Telegraph  Co.  at  82f ,  broker- 
Age  \  7o . 

18.  250  Wabash  R.  R.  at  6|,  brokerage  J%. 

19.  624  N.  J.  Central  R.  R.  at  102,  brokerage  J^. 

20.  The  total  cost  of  8700  shares  Missouri  Pacific  Is 
$185,745,  brokerage  ^7o.     What  is  their  rating? 

21.  How  are  Louisville  and  Nashville  shares  quoted  when 
4,365  shares  cost  me  $171,762.75,  brokerage  1^0%  ? 

22.  How  many  shares  of  bank  stock  at  140,  brokerage 
1%,  can  be  purchased  for  8676,243.25? 

23.  At  what  rate  must  6%  bonds  be  purchased  to  yield 
annually  5%  of  the  investment? 

Note.  The  question  is,  6%  of  the  par  /alue  is  0%  of  what  per  cent  of 
the  par  value  ;  or.  6  is  5  %  of  what  ? 

6  XlOO 
Form.    — ^ — . 

24.  At  what  rate  must  4%  bonds  be  bought  to  yield  annu- 
ally 5%  of  the  investment?  4|%  bonds  to  yield  6%  ?  7% 
bonds  to  yield  5%  ?     8%  bonds  to  yield  5|%  ? 

25o  Which  is  the  better  investment,  5%  bonds  at  95,  or 
6%  bonds  at  102^,  if  the  brokerage  on  the  first  is  U%,  and 
on  the  second  1  §  %  ? 

5        10 
Form.    961  =  19^  =  10-00  -f  19-j  =  ? 

26.  Tell  which  is  the  better  investment  in  each  of  tho  fol 
lowing  cases : 

fa)  4.V%  bonds  at  92,  brokerage  H%,  or  5^%  bonds  at 
106    brokerage  1|. 


A  PPLICA  TIONS   OF  PERCENT  A  GE.  241 

(&)  8%  bonds  at  124,  brokerage  2^%,  or  6J%  bonds  at 
110,  brokerage  1|%. 

(c)  3%  bonds  at  62,  brokerage  |%,  or  10%  bonds  at  185, 
brokerage  15%. 

27.  A  Las  a  farm  of  240  acres,  which  yields  him  an  annual 
rental  of  85]  an  acre.  A  real  estate  agent  sells  it  for  875  an 
acre,  charging  him  3%  commission.  Reserving  843.33,  A 
invests  the  net  proceeds  in  insurance  stock  at  83,  brokerage 
J%.  His  annual  income  is  increased  8203.  What  is  the 
rate  per  cent  of  the  semi-annual  dividends? 

28.  A  sells  42  100-dollar  school  bonds  bearing  7%  interest, 
at  5%  premium.  The  brokerage  was  839.  He  sent  the  net 
proceeds  to  a  Denver  broker  to  invest  in  silver-mine  stock 
at  45%  premium,  brokerage  2%.  This  stock  yielded  a  semi- 
annual dividend  of  12%.  How  much  was  his  annual  income 
increased?  "\Miat  rate  p^r  cent  did  th:  urst  broker  charge 
him  ?     "What  was  the  unused  balance  ?  ' 

29.  If  bonds  bought  J;  16|%  discount  pay  10%  on  the  in- 
vestment annually,  what  rate  oi  interest  do  they  bear? 

30.  If  stocks  bought  at  125  pay  8%  annually,  what  is  the 
rate  per  cent  of  their  semi-annual  dividends  ? 

31.  If  6%  bonds  pay  annually  4|%  of  theii  cost,  were 
they  bought  at  a  premium  or  a  discount?  Give  the  rate  per 
cent. 

32.  If  7%  bonds  yield  annually  17^%  of  their  cost,  what 
was  the  rate  of  discount? 

33.  An  electric  liglit  company,  whose  capital  stock  was 
S50,000,  earned  above  all  expenses  83,212.50  during  the  first 
half  of  the  year.  If  it  passed  8212.50  to  the  repair  fund, 
what  semi-annual  dividend  can  it  declare?  "What  will  one 
receive  who  owns  82,150  worth  of  the  stock? 

34.  A  manufacturing  company,  whose  capital  stock  was 
$250,000.  bought  new  machinery  worth  84,000.  and  leaned  an 
assessment  on  its  stockholders  to  meet  the  expenses.     "What 


242  NEW  ADVANCED  ARITHMETIC. 

was  the  rate  per  cent  of  the  assessment?     TVhat  did  A  pay, 
who  owned  $7,800  worth  of  stock? 

35.  A  milling  company,  whose  capital  stock  was  S30,000, 
declared  a  semi-annual  dividend  of  ih7c,  and  passed  ^864,50 
to  a  reserve  fund.     What  were  the  net  earnings? 

36.  How  many  shares  of  Illinois  Central  can  be  bought  for 
$7,087  at  7%  discount,  brokerage  ^%  ? 

37.  What  amount  must  be  invested  in  U.  S.  4's,  at  18J 
premium,  to  secure  an  annual  income  of  8G00,  brokerage 
being  |%  ?     What  per  cent  of  the  investment  is  the- income? 

38.  How  many  bonds  at  83^,  brokerage  J%,  can  be -pur 
chased  for  85,761.50? 

39.  How  much  must  I  invest  in  bank  stock,  paying  a 
semi-annual  dividend  of  4%,  and  selling  at  160,  brokerage 
J%,  to  yield  an  annual  income  of  §320? 

40.  Bought  75  shares  R.  R.  stock  at  28  and  sold  it  at  30^, 
brokerage  ^%  in  each  case.     What  was  the  profit? 

41.  Sold  shares  at  a  discount  of  1^%  which  I  had  pur- 
chased at  a  premium  of  ^%,  losing  §600,  brokerage  being 
J%  on  the  sale  and  |%  on  the  purchase.  How  many  were 
there? 

280.    TAXES. 

1.  In  order  to  pay  the  expenses  of  the  government,  main- 
tain public  schools  and  charitable  institutions,  build  bridges 
and  roads,  supply  towns  with  water  and  light,  and  meet 
other  expenditures  for  the  common  good,  the  State  collects 
money  from  its  citizens. 

2.  Money  collected  for  the  purposes  named  is  called  a 
Ta2. 

A  Tax  is  a  sum  of  money  levied  by  authority  of  law  upon 
persons  and  property  for  public  purposes. 


APPLICATIONS  OF  PERCENTAGE.  243 

3.  Taxes  may  be  levied  by  the  General  Government,  by 
a  State,  county,  city,  township,  or  school  district. 

Note.  The  method  of  taxation  employed  by  the  General  Government 
will  be  discussed  under  Duties  and  Imposts. 

4.  Many  States  levy  a  tax  upon  each  voter  without  re- 
gard to  the  amount  of  property  that  he  owns.  Such  a  tax 
is  called  a  Poll  Tax.  It  is  usually  a  small  amount,  rarely 
exceeding  two  dollars  a  year. 

5.  Property  may  be  either  Personal  or  Real  Estate. 
Personal  Property  consists  of  movables,  such  as  money, 

securities,  household  goods,  cattle,  etc 

Real  Estate  is  that  form  of  property  which  consists  of 
lands  and  improvements  put  upon  them. 

6.  A  Property  Tax  is  a  tax  assessed  upon  property.  It 
may  be  general  or  special. 

Note.  An  example  of  a  special  tax  is  a  paving  tax.  Give  other 
examples. 

7.  General  taxes  are  usually  assessed  and  collected  as 
follows : 

The  State  Legislature  determines  by  its  appropriation 
bills  the  amount  to  be  expended  for  State  purposes.  A 
State  officer,  usually  the  Auditor  of  Public  Accounts,  as- 
certains the  amount  of  taxable  property  in  the  State  from 
the  reports  of  certain  officers,  called  assessors,  who  are 
elected  by  the  people  in  townships  or  other  districts  of 
territory.  He  thus  discovers  what  each  dollar's  worth  of 
property  must  pay. 

The  same  thing  is  done  in  each  of  the  smaller  political 
districts,  —  the  county,  town,  etc. 

These  several  rates  of  taxation  all  find  their  way  to  the 
proper  officer,  who  calculates  the  amount  of  tax  each  indi- 
vidual and  piece  of  real  estate  must  pay,  and  puts  it  into  a 
book.  This  book  is  given  to  a  collector,  who  collects  the 
tax  and  returns  it  to  another  officer  called  a  treasurer. 


244  NEW  ADVANCED  ARITHMETIC, 

281.     PROBLEMS. 

1.  If  the  taxable  property  of  a  town  is  $568,324,  and  the 
rate  is  1^%,  what  is  the  whole  amount  of  the  tax? 

Note.  The  rate  is  often  expressed  by  the  number  of  mills  on  each 
hundred  dollars. 

2.  If  the  taxable  property  of  a  town  is  $329,864,  and  the 
tax  to  be  raised  is  $4,123.30,  what  is  the  rate  per  cent  of 
taxation?  How  many  mills  on  each  hundred  dollars?^  on 
each  dollar? 

3.  The  tax  in  a  city  is  $35,825.  The  rate  is  2^% .  What 
is  the  assessed  valuation? 

4.  The  taxable  property  in  a  city  is  $5,864,528.  The  en- 
tire tax  is  $108,809.24.  There  are  4,120  persons  subject  to 
a  poll  tax  of  $1.50  each.  What  amount  must  be  raised  by 
a  tax  on  property?  What  is  the  rate  per  cent?  How  many 
mills  on  each  hundred  dollars?  What  is  A's  tax,  whose 
personal  property  is  assessed  at  $964.80,  his  real  estate  at 
$2,160,  and  who  pays  a  poll  tax? 

5.  Calculate  the  tax  of  each  of  the  following,  the  rate 
being  If  %,  each  paying  a  poll  tax  of  $1.25  : 

(a)  Personal  property,  $736.40;  real  estate,  $2,483.00 

(&)  "               »      $1,126.84;  "         "       $5,812.40 

(c)  "  "      $5,824.90;  "         "     $12,960.25 

(d)  "  "    $12,960.65;  "        "     $42,875.49 

(e)  "  "          $439.62;  "         "       $2,974.38 

6.  Find  the  assessed  value  of  property  that  pays  $46.23 
at  1§%  ;  that  pays  $75.46  at  2|%  ;  that  pays  $86.17  at  13%  ; 
that  pays  $483.60  at  1J%,  and  includes  one  poll  tax  at 
$1.50. 

7.  The  expenses  of  a  school  district  for  one  year  are 
$2,332.40.  The  collector's  compensation  is  2%  of  the  amount 
collected.     The  taxable  property  of  the  district  is  $272,000. 


APPLICATIONS   OF  PERCENTAGE.  245 

What  is  the  amount  of  the  levy?     What  is  the  collector's 
commission  ?     What  is  the  rate  per  cent  of  taxation  ? 
Note.     Compare  "  buyiug  problems  "  in  Commission. 

8.  The  net  amount  of  a  tax  collection  is  $4,750.60.  If 
the  collector's  commission  was  2^%,  what  was  the  whole 
amount  collected?  If  the  rate  of  taxation  was  21  mills  on 
a  dollar,  what  was  the  amount  of  taxable  property  ? 

9.  Net  collection,  $.3,824.80;  collector's  compensation, 
2%.  What  was  the  amount  of  his  commission?  What 
amount  did  he  collect? 

10.  Collector's  commission,  $348.80;  rate  of  commission, 
1|%.     What  amount  should  he  turn  over? 

11.  Amount  collected,  $53,860.40;  amount  not  collected, 
$1,620.12;  collector's  commission,  2%;  rate,  13  mills  on  a 
dollar.     What  was  the  assessment? 

12.  The  State  tax  is  61  mills  on  $100.  The  local  tax  is 
18  mills  on  $1.00.  What  is  the  total  tax  on  property  as- 
sessed at  $1,560? 

13.  The  assessed  valuation  being  $1,324,000,  and  the  rate 
of  taxation  being  1|%,  what  is  the  collector's  commission  at 
2%  if  he  collect  all  but  $625  of  the  tax? 

14.  The  rate  of  taxation  is  8^  mills  on  the  dollar.  What 
is  the  acre  valuation  of  a  farm  of  320  acres  upon  which  the 
tax  is  $163.20? 

282.  UNITED  STATES  REVENUE. 

1.  The  money  necessary  to  meet  the  expenses  of  the 
United  States  Government  is  chiefly  derived  from  a  tax 
imposed  upon  goods  imported  from  foreign  countries,  called 
Customs  or  Duties,  and  from  a  tax  imposed  upon  certain 
wticles  manufactured  within  the  country,  called  Excise 
Taxes. 

2.  Duties  are  collected  at  certain  cities  designated  by 
kw,  and  called  Ports  of  Eatry.     Each  port  of  entry  oon- 

17A 


246  NEW  ADVANCED  ARITHMETIC. 

tains  a  custom-house.  The  collection  of  duties  is  under  the 
direction  of  au  officer  called  Collector  of  the  Port,  appointed 
by  the  President  and  confirmed  by  the  Senate  of  the  United 
States. 

3.  These  Duties  are  either  Specific  or  Ad  Valorem. 
Specific  duties  are  duties  assessed  upon  imported  goods 

with  reference  to  their  quantity,  and  not  to  their  value. 
Such  duties  are  not  computed  by  percentage. 

Ad  valorem  duties  are  duties  assessed  upon  imported 
goods  with  reference  to  their  cost  in  the  countries  from 
which  they  are  brought. 

4.  Importers  submit  an  invoice  which  shows  the  cost 
of  each  article  imported  in  the  country  where  it  was 
pm-chased. 

5.  A  second  source  of  revenue  to  the  United  States  Gov- 
ernment is  a  tax  imposed  upon  spirituous  liquors  manufac- 
tured within  the  country. 

6.  A  third  source  of  revenue  is  the  sale  of  public  lands. 

7.  On  certain  kinds  of  imports  an  allowance  called  tare 
is  deducted  for  the  weight  of  the  cases  containing  the  goods, 
for  breakage,  leakage,  etc. 

283.    PROBLEMS. 

1.  A  dealer  imported  65  watches  invoiced  at  828.  What 
was  the  duty  at  25  %  ? 

2.  What  is  the  duty  at  15%  on  15  oil  paintings,  aver- 
aging S2,350  each,  and  on  seven  pieces  of  statuary,  averaging 
SI, 750  each? 

3.  AVhat  is  the  duty  on  1  gross  of  brushes  at  40%,  in- 
voiced at  S7.85  a  dozen? 

4.  The  duty  on  lace  is  60%.  What  was  the  number  of 
yards  in  an  importation  upon  which  the  duty  was  $624,  the 
lace  being  invoiced  at  80  cents  a  yard  ? 


APPLICATIONS   OF  PERCENTAGE.  247 

5.  Wbat  is  the  rate  of  duty  upon  musical  instruments, 
■when  the  duty  on  the  following  importation  amounted  to 
61,679.40? 

24  violins,  invoiced  at  842  ? 

7  pianos  "  "  8324? 

12  flutes  "  "838? 

6.  Find  the  dut}^  on  the  following  importation : 

81,575  worth  of  sponges  at  20%. 

285  pounds  of  spices  at  4  cents  a  pound. 

85,000  worth  of  leather  at  10%. 

7.  What  is  the  duty  on 

2,650  pounds  of  cheese,  duty  5  cents  a  pound? 
3,650  pounds  of  tin  plate,  duty  2-^  cents  a  pound? 
4,200  pounds  of  rice,  duty  2  cents  a  pound? 

8.  What  is  the  duty  at  2  cents  a  pound  on  300  cases  of 
starch,  each  containing  50  pounds? 

9.  What  is  the  duty  on  360  j'ards  of  silk,  invoiced  at 
81.12,  at  50%  ad  valorem? 

10.  Find  the  duty  on  1,200  j-ards  of  Irish  linen,  invoiced 
at  35  cents,  at  32%  ad  valorem? 

11.  What  is  the  duty  on  one  gross  toilet  bottles,  invoiced 
at  83  a  dozen,  at  33  J  %  ad  valorem,  with  an  allowance  of 
5  %  for  breakage  ? 

12.  What  is  the  specific  duty  on  50  bags  of  rice,  each  con- 
taining 150  pounds,  at  2  cents  a  pound,  the  tare  being  3%  ? 

13.  What  is  the  specific  duty  on  600  pounds  almonds, 
at  5  cents  a  pound? 

14.  Find  the  duty  on  624  yards  Brussels  carpet  (|  yard 
wide),  invoiced  at  58  cents,  the  ad  valorem  duty  being  40%, 
and  the  specific  duty  28  cents  a  square  3'ard. 

15.  What  is  the  duty  on  8  dozen  watch  movements,  in- 
voiced at  86.80  each,  the  duty  being  25%  ? 

16.  Find  the  duty  on  24,000  pounds  of  bituminous  coal 
at  45  cents  for  a  long  ton. 


248  NEW  ADVANCED  ARITHMETIC. 

17.  Find  duty  on  6  lb.  7  oz.  8  pwt.  of  a  drug  which  cost 
15  cents  an  ounce,  the  duty  being  33%  ad  valorem. 

18.  Find  duty  on  4  dozen  watch-cases  averaging  2  oz.  12 
pwt.  18  gr.,  invoiced  at  $1,36  an  ounce,  the  duty  being  25%. 

19.  Find  duty  on  24  cases  cloth,  each  containing  36 
yards,  invoiced  at  $1.38,  duty  being  55%. 

20.  Find  duty  on  368  pounds  cork,  duty  being  15  cents 
a  pound,  and  tare  11%. 

284.    INSURANCE. 

1.  Houses  are  liable  to  be  destroyed  by  fire  or  tornadoes, 
ships  to  be  wrecked,  and  persons  upon  whom  others  are 
dependent  to  be  injured  or  to  die. 

2.  Because  of  these  facts  companies  are  organized  that, 
for  a  certain  amount,  agree  to  pay  for  property  thus  de- 
stroyed, to  allow  a  person  a  stated  amount  if  injured,  or  to 
pay  to  his  heirs  a  fixed  sum  in  case  of  his  death.  Such 
companies  are  called  Insurance  Companies. 

3.  Insurance  is  security  against  financial  loss  on  account 
of  the  destruction  of  property,  or  by  the  injury  or  death  of 
a  person. 

4.  The  different  kinds  of  insurance  receive  their  names 
from  the  dangers  against  which  they  offer  insurance  : 

5.  The  common  forms  of  insurance  are : 

(a)  Fire  and  Lightning.  (d)  Accident. 

(b)  Tornado.     ^  (e)   Life. 

(c)  Marine. 

6.  The  company  taking  the  risk  is  called  the  Insurer,  or 
Underwriter. 

7.  The  property  or  persoti,  upon  whom  the  risk  is  taken, 
is  insured. 

8.  A  Policy  is  a  contract  made  between  the  insurer  and 
the  person  securing  the  insurance. 


APPLICATIONS   OF  PERCENTAGE. 


249 


9.  The  Premium  is  the  amount  paid  to  the  insurer  for 
assuming  the  risk. 

10.  The  underwriters  assume  the  risk  for  the  length  of 
time  specified  in  the  pohcy. 

11.  In  Property  Insurance  the  premium  is  estimated  at  a 
certain  per  cent  of  the  amount  of  the  risk  for  a  specified 
time. 

12.  In  Accident  Insurance  the  premium  is  estimated  at  a 
(•ertain  amount  for  tlie  protection  afforded  for  a  given  time. 

13.  In  Life  Insurance  the  premium  is  estimated  at  a  cer- 
tain amount  per  year  for  each  thousand  dollars  of  insurance, 
and  varies  with  the  age  of  the  insured. 


285.     PROPERTY    INSURANCE. 
PROBLEMS. 

1.  What  is  the  premium  on  the  following  policies : 

((')  $2,500  for  3  years,  at  ^%  a  year. 
(6)  S3,650    ''    5         ^'  \% 

(c)  $4,680    "    2         "  2%        " 

(d)  $8,250    "     1   year,  at  1|%        " 

2.  Find  the  annuo'  rate  per  cent  of  the  premium  on  the 
following  policies : 

Premium. 


Amount. 

(a)     $3,600 
(&)      $4,250 

(c)  $8,400 

(d)  $24,600 


Time. 
5  5'ears. 
4       " 
3      " 
2       " 


$68 

$315 

$1,230 

3.    Find   the    amount  insured    in    each  of   the   following 
policies : 

Premium.  Annual  Rate.  Time, 

(a)  $68.24  *%  1  year. 

(&)  $750  11%  3  years, 

(o)   $872.73  H%  5      " 

(d)   $463.40  2i%  2      " 


250  NEW  ADVANCED  ARITHMETIC. 

4.  A  <?chooIhouse  worth  $32,400  is  insured  for  f  of  its 
value  at  ^%  annually  for  3  years.     Find  the  premium. 

5.  A  ship  worth  852,000  was  insured  for  %  of  its  value  at 
2J%.  The  cargo,  worth  $8,640,  was  insured  for  §  of  its 
value  at  3%.     What  was  the  whole  premium? 

6.  Paid  $290.19  for  insuring  a  cargo  worth  $13,656.  What 
was  the  rate  per  cent  of  the  premium? 

7.  My  house  being  destroyed  by  fire,  I  received  from  the 
underwriters  $2,760,  which  was  §  of  the  amount  of  the  policy. 
Insurance  was  |  of  value  of  house  ;  the  premium  was  $66.24. 
Find  the  rate,  and  the  value  of  house. 

8.  A  factory  worth  $75,000  was  insured  for  4  of  its  value 
by  an  insurance  company  at  2^% .  Not  wishing  to  carry  the 
entire  risk,  the  company  reinsured  \  of  the  risk  at  1|% ,  J  of 
the  risk  at  2^%,  and  i  of  the  remainder  at  2%.  What  was 
the  amount  of  the  premium  remaining  for  the  first  company  ? 

LIFE    INSURANCE. 

1.  Find  annual  premium  of  a  policy  of  $8,000  at  $17.54  a 
thousand. 

2.  Find  annual  premium  on  an  endowment  policy  of 
$10,000  at  $85.60  a  thousand. 

3.  In  5  years  I  have  paid  premiums  amounting  to  $472.50 
on  my  life  policy  of  $5,000.  What  is  the  annual  premium 
for  $1,000? 

4.  For  8  years  I  have  made  bi-monthly  payment  of  $7.86 
as  premiums  on  my  policy  in  a  Mutual  Insurance  Company. 
What  is  the  aggregate?  The  annual  premium  is  $15.72  per 
$1,000.     What  is  the  amount  of  my  policy? 

5.  A  person  carried  a  policy  for  $6,000  for  9  years,  pay- 
ing annual  premiums  of  $31.80  per  $1,000.  At  the  end  of 
that  time  he  surrendered  his  policy  for  23%  of  what  he  had 
paid.     What  did  he  receive?   . 


APPLICATIONS   OF  PERCENTAGE.  251 

6.  A  person  insured  his  life  for  87,000  at  an  annual  pre- 
mium of  836.40  per  81,000.  After  making  7  payments  he 
died.  How  much  more  did  his  heirs  receive  than  he  had 
paid  ? 

7.  A  man  carried  a  20-3-ear  endowment  policy  for  82,000 
at  an  annual  premium  of  644.70  per  81,000.  His  dividends 
aggregated  165%  of  the  premiums.  How  much  more  did  he 
receive  at  the  maturity  of  the  policy  than  he  had  paid  ? 

8.  If  I  pay  an  annual  premium  of  869.80  per  81,000  on  a 
10-year  endowment  policy  of  84,500,  what  amount  has  been 
paid  at  the  maturity  of  the  policy? 

286.      MISCELLANEOUS    PROBLEMS. 

1.  Change  yij  to  a  decimal  fraction. 

2.  When  it  is  45  minutes  past  8  a.  m.  at  Buffalo,  78°  55' 
W..  what  is  the  time  at  Salt  Lake  City,  112°  6'  W.  ? 

—  g   X  O2  ^   "I"    8 

3.  Reduce  03  ^  ■■  7  X  jr y 

4.  Bought  36,824  bushels  of  oats  at  18|  cents  and  sold 
them  at  a  gain  of  12^% .     "What  did  they  bring? 

5.  Settled  a  bill  of  8436.50  at  a  discount  of  16  and  5. 
What  was  the  amount  paid  ? 

6.  Find  tax  on  a  real  estate  assessment  of  85,640  and  a 
personal  property  assessment  of  83,824,  at  15  mills  on  the 
dollar. 

7.  What  is  the  commission,  at  2^%,  on  the  sale  of  320 
acres  of  land  at  872.50  an  acre? 

8.  Find  the  premium  on  a  policy  insuring  a  house  worth 

84.860  for  f  of  its  value  at  1^% . 

9.  Divide  .00864  by  2.7. 

10.  Bought  a  stock  of  goods  for  88,645  and  sold  them  for 
89,748.     What  per  cent  did  I  gain  ?     (Approximate.) 


252  NEW  ADVANCED  ARITHMETIC. 

11.  How  many  silver  table-spoons,  each  weighing  1  oz.  7 
pwt.,  in  a  package  weighing  2  lb.  8  oz.  8  pwt.  ? 

12.  Find  the  cost  of  8  SlOO-bonds,  bearing  4^%  interest, 
which  yield  an  annual  income  of  6^%  of  the  investment. 

13.  The  difference  of  time  of  two  places  is  3  hr.  40  min. 
30  sec.  If  the  place  having  the  later  time  is  iu  15°  24'  12" 
east  Ion.,  what  is  the  longitude  of  the  other? 

14.  What  is  the  duty  on  drugs  weighing  6  lb.  8  oz.  5  dr. 
2  sc,  invoiced  at  12  cents  a  dram,  the  duty  being  35%  ? 

15.  "What  is  the  area  of  a  circular  pond,  whose  diarneter  Is 
42  rods? 

16.  A  commission  merchant  sold  for  me  8,624  bushels  of 
corn  at  28i  cents.  After  deducting  his  commission  at  3^%, 
and  paying  charges  amounting  to  S38.75,  he  invested  the 
remainder  in  oats  at  16^  cents  a  bushel,  first  withdrawing 
his  commission  at  2^  %.     How  many  bushels  did  he  buy? 

17.  Find  the  cost  at  S5.50  a  cord  of  a  pile  of  wood  8  feet 
high,  4  feet  wide,  and  36  feet  long. 

18.  Sold  a  lot  of  goods  for  S650.40.  After  paying  2^% 
commission  to  the  auctioneer,  I  find  that  I  have  made  a  net 
gain  of  24% .     What  did  they  cost  me? 

19.  AVhat  is  the  capacity  in  barrels  (3H  gallons)  of  a 
cylindrical  cistern  whose  diameter  is  10  feet  and  depth  12 
feet?     (tt  =  2^.     Employ  cancellation.) 

20.  What  is  the  rate  of  income  on  4%  bonds  bought  at 
16%  discount? 

21.  How  far  will  a  train  go  in  7  hours,  if  its  average 
speed  is  36  mi.  124  rd.  4  yd.  an  hour? 

22.  What  is  the  area  of  a  triangular  piece  of  land  whose 
base  is  124  rods  and  whose  altitude  is  236  rods? 

23.  The  valuation  of  a  town  is  82,624,800.  The  tax 
levy  is  §32,810.     Give  the  rate  in  mills. 


APPLICATIONS   OF  PERCENTAGE.  253 

24.  What  is  gained  by  buying  24  shares  of  Illinois  Central 
at8|%  discount  and  selling  them  at  93^,  brokerage  \%  in 
each  case  ? 

25.  A  owns  f  of  a  mill,  and  sells  |  of  his  share  to  B,  who 
then  owns  ^  of  the  mill.  What  part  of  the  mill  did  he  own 
before  his  purchase?     B's  share  is  now  what  part  of  A's? 

26.  4^  inches  is  what  decimal  of  a  rod? 

27.  What  is  the  value  of  8  sUls,  each  10'' x  10",  and  16 
feet  long,  at  $19.50  a  thousand  feet? 

28.  A  bill,  having  been  discounted  25  and  5,  amounts  to 
$855.     What  is  its  face? 

Note.     Employ  cancello,tion  when  possible. 

29.  At  what  advance  must  goods  be  marked  so  that  the 
merchant  may  discount  the  marked  price  20  aud  5,  and  still 
make  14%? 

30.  Find  premium  on  a  policy  insuring  a  house  worth 
$6,000  for  I  of  its  value,  for  a  term  of  5  years,  the  annual 
rate  being  ^%. 

31.  If  15  men  working  8  hours  a  day  for  25  days  can  do 
a  piece  of  work,  how  many  hours  a  day  must  18  men  work 
to  complete  the  same  job  in  20  days  ? 

32.  Divide  133574  by  .000329. 

33.  Find  the  cost  of  lumber  and  posts  for  fencing  the 
E.  I  of  S.  E.  ^  of  a  section  with  a  4-board  fence,  posts 
costing  18^  cents,  and  lumber  $19.25  per  thousand. 

34.  How  many  bushels  of  oats  will  a  bin  contain  that  is 
10  feet  high,  8  feet  wide,  and  32  feet  long? 

35.  How  many  shares  at  83,  brokerage  Yfo,  can  be  pur- 
chased for  $4,571.88? 

36.  Sugar  was  selling  at  20  pounds  for  a  dollar.  The 
price  advanced  16f%.  How  many  pounds  less  could  then 
be  bouo;ht  for  a  dollar? 


254  NEW  ADVANCED  ARITHMETIC. 

37.  Sent  to  a  commission  merchant  $894  to  be  invested 
after  withdrawing  his  commission  at  3%.  What  is  the 
amount  of  his  commission  ? 

38.  What  must  be  the  depth  of  a  cylindrical  vessel  whose 
diameter  is  6  inches,  to  hold  a  gallon  ? 

39.  A  can  do  a  piece  of  work  in  7  days,  B  in  8  days,  and 
C  in  9  days.     In  what  time  can  they  do  it,  working  together? 

40.  If  A  worked  3  days  and  B  4  days,  in  what  time 
could  C  finish  the  work? 

41.  ^  of  A's  money  equals  f  of  B's.  If  A  has  865  more 
than  B,  how  nmch  has  each? 

42.  Bought  a  house  for  $5,680.  After  renting  it  for  one 
year  at  S41  a  month,  and  spending  $121  for  taxes  and 
repairs,  I  sold  it  for  $6,000.    What  is  the  per  cent  of  gain? 

43.  What  is  the  commission  on  the  following  sales? 
3,640  bushels  of  oats  at  18  cents.     Commission  2%. 
2,500  bushels  of  wheat  at  53i^  cents.     Commission  2\%. 
3,600  bushels  of  corn  at  21h  cents.     Commission  2^%. 

44.  J  of  jV  is  what  per  cent  of  g  ^  3  ? 

45.  How  many  cubic  yards  of  earth  in  150  loads  averraging 
24  cu.  ft.  1,200  cu.  in.? 

46.  .023907-^.0001839=? 

47.  What  is  the  area  of  a  circular  field  whose  diameter  is 
45.12  rods? 

48.  Find  the  cost  of  the  Brussels  carpet  for  a  room 
20'  X  24',  the  strips  running  the  long  way,  8  inches  being 
lost  on  each  strip  in  matching,  and  costing  $1.12  a  yard. 

49.  A  wood-house  is  16'  x  16'.  If  filled  with  wood  to  a 
height  of  7  feet,  what  is  the  wood  worth  at  $5.50  a  cord? 

50.  By  selling  an  article  for  $565,  I  gain  13%.  AVliat 
did  it  cost  me? 

51.  Find  the  value  of  a  pile  of  cordwood  7  feet  high  and 
36  feet  long,  at  $5.25  a  cord. 


APPLICATIONS   OF  PERCENTAGE.  255 

287.     INTEREST. 

1.  Many  persDns  have  occasion  to  use  money  when  they 
do  not  have  it  at  hand.  If  they  are  responsible,  they  can 
obtain  the  money  from  others  by  guaranteeing  its  return  and 
by  agreeing  to  pay  a  specified  sum  for  such  service.  Such 
persons  are  called  borrowers. 

2.  The  amount  paid  for  such  service  is  called  Interest. 
Interest  is  compensation  for  the  use  of  money. 

Note.  Mouev  paid  for  labor  is  usually  called  wages ;  when  paid  for 
the  use  of  houses  and  lands,  it  is  called  rent,  etc. 

3.  The  quantity  of  money  upon  which  interest  is  paid  is 
the  Principal. 

4.  The  Amount  is  the  sum  of  the  principal  and  interest. 

5.  Wages  and  rent  are  usually  estimated  at  a  specified 
amount  for  a  day,  a  month,  or  a  year.  Time,  consequently, 
is  an  essential  element  in  calculating  such  compensation. 
Similarly,  the  longer  the  time  that  one  uses  the  money  of 
another,  the  larger  should  be  the  compensation. 

6.  The  ordinary  unit  of  time  for  estimating  interest  is 
one  year.  The  Rate  of  Interest  is  a  specified  number  of 
hundredths  of  the  principal,  as  a  compensation  for  its  use 
for  one  year;  hence,  Interest  is  one  of  the  applications  of 
percentage. 

7.  When  money  is  loaned  at  6%,  it  is  understood  that  the 
compensation  is  6  hundredths  of  the  principal,  for  its  use  for 
one  year,  unless  otherwise  specified. 

8.  Interest  is  one  of  the  most  common  of  the  applications 
of  percentage.  Since  the  time  for  which  money  is  borrowed 
may  vary  from  a  few  days  to  many  months  or  years,  it  is 
the  most  difficult  of  these  applications. 

9.  Remember  that  problems  in  interest  involve  : 

(1)  A  sum  upon  which  interest  is  calculated ;  (2)  A  rate 
of  interest,  —  a  number  of  hundredths  of  that  sum  for  its  use 
for  a  unit  of  time,  usually  a  year ;  (3)  A  specified  time  for 
which  the  given  sum  is  to  be  loaned. 


256  NEW  ADVANCED  ARITHMETIC. 

288.     ORAL    PROBLEMS. 

1.  What  is  the  interest  on  $800  for  one  year  at  6%  ?  at 
8%?     at9%?     at4%?     at  12%  ?     at  10%?     at7%? 

2.  What  is  the  interest  on  $600  for  2  years  at  3  %  ?  at 
5%?     at  71%?     at4J%? 

Note.     First  find  the  interest  for  one  year. 

3.  Substitute  3  years  for  1  year  in  the  first  problem. 

4.  What  is  the  interest  on  $900  for  2  years  and  6  months 
(2^  years)  at  6%  ?  at  4%  ?  at  3%  ?  at  8%  ? 

5.  What  is  the  interest  on  S400  for  3  years,  3  months,  at 
6%?  4^?  5%?  8%?  3%?  10%? 

289.  There  are  many  methods  of  calculating  interest. 
Persons  whose  business  requires  a  considerable  amount  of 
such  work  supply  themselves  with  books  containing  "  In- 
terest Tables."  With  their  assistance  it  is  very  easy  to 
perform  any  problem  in  interest. 

For  the  ordinary  person  one  good  method  is  all  that  is 
needed.  Several  methods  will  be  presented,  but  proficiency 
in  two  is  recommended.  These  are  the  "  General  Method  " 
and  the  first  "  Six  Per  Cent  Method." 

290.    WRITTEN  PROBLEMS. 

General  Method. 

Illustrative  Problem.  What  is  the  interest  on  $750  for 
2yr.  7  mo.  15  d.  at  6%? 

FORM. 

$750 

.06 
$15.00    =  interest  for  1  yr. 

2 


6mo.  =  ^yr.  22., 50    = 

1  mo.  =  1^  of  6  mo.       3.75    = 
15  d.  =  I  of  1  mo.        1.875  = 


125  = 


2  yr. 
6  mo. 

1  mo. 
15  d. 

2  yr.  7  mo.  15  d. 


APPLICATIONS   OF  PERCENTAGE. 


257 


RULE. 
For  finding  interest  by  the  general  method: 

1.  Find  the  interest  for  one  year  nt  the  given  rate, 
and  multiply  the  result  by  the  number  of  years. 

2.  Separate  the  niontlis  into  divisors  of  12,  and  find 
such  a  part  of  the  interest  for  one  year  as  each  of  these 
divisors  is  of  12. 

Note.  It  will  be  found  more  convenient  sometimes  to  compare  some 
of  the  divisors  with  others,  as  in  the  illustrative  problem. 

3.  Separate  the  flays  into  divisors  of  SO.  Find  the 
interest  for  one  month,  and  proceed  as  in  (2). 

Note.  An  arrangement  of  the  days  can  be  made  sometimes  similar 
to  that  of  the  mouths,  as  suggested  in  the  preceding  note. 

291.  In  the  following  problems  follow  the  above  method. 
Count  30  days  for  a  m  th.  Reject  all  results  below  tenths 
of  mills.     Find  the  amount  in  each  case. 


Principal 


Time. 


1. 

$480 

4% 

3:y 

r.  5  mo.  10  d. 

2. 

$650.40 

5% 

2 

'     7     ' 

15  " 

3. 

$864  36 

6% 

4 

'    8    ' 

6  " 

4. 

$127.5.86 

8% 

1 

'    9     ' 

6  '* 

5. 

$1464.29 

10% 

2 

"    1    ' 

'      3  " 

6. 

$2580.47 

7% 

3 

'     2     ' 

10  " 

7. 

$3600 

H% 

5 

'  10     ' 

15  " 

8. 

$4580 

b\% 

4 

'     6    ' 

2   " 

9. 

$5000 

%\% 

5 

'    3     ' 

1    " 

10. 

$6840.75 

11% 

6 

'  11     ' 

19   " 

11. 

$490.92 

8% 

1 

'    3     ' 

6  " 

12. 

$3794.08 

9% 

2 

.     4     ' 

7  " 

13. 

$892.45 

5% 

3 

'     4     ' 

8  " 

14. 

$1234.16 

6% 

2 

'     3     ' 

12   " 

Note.     Correct   solutions   may  differ  by  a  few  cents   in   the   results 
because  of  different  divisions  of  the  time. 


258 


NEW  ADVANCED  ARITHMETIC. 


Principal. 

Rate. 

Time. 

15. 

85871.48 

7% 

4  yr.   1  mo.  20  d. 

16. 

S«9.26 

3% 

6    "    8    • 

'    25  " 

17. 

613.51 

5J% 

5    "  10    ' 

•    16  " 

18. 

Si. 05 

H% 

8   "    2    ' 

'    24  " 

19. 

S10874.80 

4% 

5    ' 

'    10  » 

20. 

S916.15 

8% 

9    ' 

'    18  " 

21. 

S1371.49 

5% 

1    "  11    ' 

'    22  '* 

22. 

81641.04 

9% 

3    "    2    ' 

'    29   " 

23. 

83200 

10% 

2  •"    7    ' 

'      1   -' 

24. 

893.94 

12% 

10    ' 

'      4  " 

25. 

816921.72 

3i% 

3    ' 

'    11  " 

26. 

8«79.40 

4% 

2    ' 

'    4    • 

'    10  " 

27. 

8783.12 

5i% 

3    ' 

'    S     ' 

'    1  o  " 

28. 

81864.60 

7% 

1    ' 

'     7     ' 

'    18  " 

29. 

81460.20 

7% 

4    ' 

'  10    ' 

'    21   " 

30. 

821.25 

6% 

4    ' 

'    1     ' 

t      2  '' 

31. 

8583.40 

6% 

5    ' 

'    3    ' 

'    16  " 

32. 

81560 

7i% 

2    ' 

'  11     ' 

'    22  " 

33. 

8830.80 

7h7c 

8    ' 

'    5    ' 

i    25  " 

34. 

898.07 

41% 

3    ' 

'    6    ' 

'    28  « 

35. 

82654.90 

^% 

6    ' 

'    7    ' 

'      7  " 

36. 

83371.10 

8% 

5    ' 

'    5     ' 

'    11   '* 

37. 

8640 

Sh7o 

4    ' 

'    4    ' 

i      4  u 

38. 

8756.50 

31% 

5    ' 

'    3    ' 

'      8  " 

39. 

8833.12 

4^% 

7    ' 

'    6    ' 

'    13  " 

40. 

8794.80 

41% 

8    ' 

'    8    ' 

'      7  " 

41. 

81026.25 

5% 

9    ' 

'     1     ' 

'    14  " 

42. 

81230 

5% 

4    ' 

'    2    ' 

'    16  " 

43. 

81426.80 

5i% 

3    ' 

'  10    ' 

'      2  " 

44. 

81862.60 

6% 

7    ' 

'    5    ' 

'    19  " 

45. 

82045 

6% 

6   ' 

'    9    ' 

'    21   " 

APPLICATIONS   OF  PERCENTAGE.  259 


Principal. 

Rate. 

Time. 

46. 

$3870 

6i% 

2y. 

11 

mo 

20  d. 

47. 

S4312 

61% 

3    '• 

11 

" 

25  " 

48. 

$5680 

7% 

5    " 

10 

u 

22  " 

49. 

$6875 

7% 

6 

t( 

24  " 

50. 

§5892 

7% 

3 

(( 

22  " 

51. 

$66.40 

7% 

24  " 

52. 

$83.70 

7% 

18  " 

53. 

$168.10 

8% 

2   " 

10 

u 

29  " 

54. 

$234.90 

8% 

5    " 

24  '* 

55. 

$361.12 

9% 

7 

u 

16  " 

56. 

$432.90 

10% 

11 

(( 

22  " 

57. 

$563.08 

10% 

24  » 

58. 

$839.16 

10% 

20  " 

59. 

$748.90 

12% 

26  " 

60  0 

$2126.40 

12% 

12  " 

292.    Six  P6r  Cent  Methods. 

(a)  The  Method  bt  Aliquot  Parts. 

Note.  An  "  aliquot  part  "  of  a  number,  as  here  used,  is  a  part  of  a 
number  which  is  expressed  by  a  fraction  whose  numerator  is  I. 

1.  Siuce  any  principal  which  bears  interest  at  6%  per 
annum  gains  j§o  of  itself  in  a  year,  it  gains  y-Jo  of  itself 
in  2  months,  and  doubles  itself  in  200  months. 

What  part  of  itself  will  it  gain  in  100  months?  in  50? 
40?  20?  10?  2?  33^?  66|? 

2.  In  computing  interest,  1  month  =  30  days;  hence,  2 
months  =;  60  days ;  hence,  the  interest  on  any  principal  for 
60  days  at  6%  is  jl^  of  that  principal. 

Since  6  days  is  yV  of  60  days,  the  interest  for  6  days  is 
iV  of  TWO  =  tttVtt  of  ttie  principal. 


260 


NEW  ADVANCED  ARITHMETIC. 


3.    FACTS   TO   BE   REMEMBERED. 

Any  principal  at  6% 

(1)  Doubles  itself  in  200  months. 

(2)  Gains  ji^  of  itself  in  60  days. 

(3)  Gains   yoVn  °^  itself  in  6   days. 

4.    Illustrative  Problem.     Plnd  the  interest  on   $860  for 
4  yr.  7  mo.  24  d.  at  6%. 

Form. 


4  yr. 
50  mo.  =  1  of  200  mo. 

mo.  =  OD  mo. 

$860 

$215         =  interest  for  50  mo 

5    "     =  -L  of    50    " 

20  d.     =  J  of    60  d. 

4  "      =  1  of    20  " 

21.50    =        "         "5    " 

2.866=        "        "    20  d. 

.573  =        "        "4  " 

$239.94 
Solve  the  following  by  the  above  method : 


Note 


Note 


2[ 

)3.      PROBLEMS. 

1. 

Principal. 
$540 

3yr. 

Time. 

3  mo.  15  d. 

Rate 

6% 

2. 

$650 

4  " 

8    "     20  " 

3. 

$1240.50 

6  " 

9    "     12  " 

4. 

$980.40 

5  " 

6    '^     20  " 

.     5 

yr.  6  mo.  20  d. 

=  66|  mc 

.  =  ^  of  200  mo. 

5. 

$2450.72 

8  yr. 

4  mo. 

6. 

$6000 

2   '' 

9    "     10  d. 

.     2 

yr.  9  mo.  10  d. 

=  33|  mo 

.  =  ^  of  200  mo. 

7. 

$8450 

8  yr. 

10  mo.  26  d. 

8. 

$1750.86 

3  " 

11    "     29  " 

9. 

$5826.25 

4  " 

9    "     19  " 

10. 

$349.46 

5  '■' 

1     u         1   u 

11. 

$850 

6  " 

2    "     13  " 

8%. 

APPLICATIOXS   OF  PERCENTAGE.  261 

KoTE.  First  find  the  interest  at  6%,  then  divide  the  result  by  6  and 
multiply  the  (juotieut  by  the  number  of  per  cent.  Why  ?  Or,  add  to  the 
result  i  of  itself.     Why  ? 

12.  Prin.,Sll48.12.     Time,  4  yr.  8  mo.  U  d.     Rate,  9%. 

13.  Solve  1,  2,  3  at  TTe. 

14.  Solve  4,  5,  C  at  5%. 

15.  Solve  7,  8  at4-|-%. 

16.  Solve  9,  10  at  10^^. 

For  further  practice  in  this  method  use  problems  under 
Art.  291. 

294.     {h)  The  Method  by  First  Finding  the  Interest 
ON  §1.00. 
Prove  the  following  statements  : 

1.  The  interest  on  Sl.OO  at  6%  is  ^  a  cent  a  month,  and 
Jjj  of  a  cent  for  G  days. 

Method.  Express  the  time  in  months  and  days.  The  m- 
terest  on  Si. 00  at  6%,  for  any  number  of  months  and  days, 
is  \  as  many  cents  as  there  are  months,  and  ^  as  many  mills 
as  there  are  days. 

Illustrative  Problem. 
Principal,  3560.25.     Time,  4  yr.  7  mo.  15  d.     Rate,  G%. 
4  yr.  7  mo.  =  55  mo. 
The  interest  on  Si  for  55  mo.  =  SO. 275    cents. 

"         "         "   15  d.  =      .0025      " 

"  "  "  55  mo.  15  d.  =  SO. 2775  cents. 

The  interest  on  S560.25  is  560.25  x  SO. 2775  cents. 
SO. 2775  cents. 

560.25 
.013875 
.05550 
16.650 
138.75 


18^  S155.469375 


262  NEW  ADVANCED  ARITHMETIC. 

Note.  Reject  results  below  mills.  Where  more  convenient,  use  first 
result  as  multiplier. 

Perform  Problems  1-11,  Art.  293,  by  this  method. 

295.     Method  by  changing  Rate   Per  Cent  for  One  Year 
to  Rate  Per  Cent  for  the  Whole  Time. 

1.  Since  in  calculating  interest  it  is  common  to  consider 
360  days  as  equal  to  a  year, 

Reduce  the  years  to  mouths,  and  to  this  result  add  the 
number  of  months. 

Thus :  3  yr.  4  mo.  =  40  mo.  etc. 

2.  Since  there  are  30  days  in  a  month,  3  days  equal  ^  of 
a  month  ;  hence,  dividing  the  days  by  3  reduces  them  to 
tenths  of  a  month. 

3  yr.  7  mo.  18  d.  =  43.  6  mo. ;  5  yr.  9  mo.  23  d.  =  69. 7§  mo. 

3.  Express  the  following  periods  of  time  as  mouths  and 
tenths  : 

(1)  2  yr.     2  mo.     7  d.  (4)  5  yr.    8  mo.  25  d. 

(2)  3  "      5    "       9  "  (5)  8  "    10    "     29  " 

(3)  4  "    11    ''     16  " 

4.  12%  a  year  is  1%  a  month  ;  hence,  if  the  time  be  ex- 
pressed in  months  and  tenths  of  months,  the  result  will  be 
the  number  of  hundredths  of  the  principal  which  the  interest 
would  be  at  12%  per  annum.  To  find  the  corresponding 
number  for  any  other  rate  per  cent,  divide  this  result  by  12 
and  multiply  the  quotient  by  the  number  of  ones  in  the  rate 
per  cent. 

^  5.  The  interest  upon  any  principal  is  then  found  by  finding 
as  many  hundredths  of  it  as  the  preceding  result  expresses. 

Illustrative  Prohle.m.  Find  the  interest  on  $820  for  4  yr. 
8  mo.  12  d.  at  6%.  4  yr.  8  mo.  =  56  mo. ;  12  d.  =  .4  mo. 
The  time  =  56.4  mo.  The  interest  is  56.4%  of  the  principal 
at  12%,  4.7%  at  1%,  and  28.2%  at  6%. 

Note.  It  is  sometimes  more  convenient  to  find  the  interest  at  12%  and 
then  take  such  a  part  of  the  result  as  the  givpu  rate  is  of  12% . 


APPLICATIONS   OF  PERCENTAGE.  263 

Solve  Problems  1-10,  Art.  291,  by  this  method. 
296.    Keview  method  of  finding  time  between  two  date?  in 
Compound  Denominate  Numbers. 

Note.  The  fullowiug  method  is  often  employed  :  Write  for  the  minu- 
end the  year,  number  of  inontL,  number  of  day  in  the  month  of  the  later 
date ;  and  for  the  subtraliend,  the  corresponding  numbers  of  the  earlier 
date.  Proceed  as  in  subtraction  of  co-i.^'ound  numbers,  counting  30  days 
for  a  mouth . 

Illustration. 

Later  date,     July  12,  1891  —  1891  7  12 
Earlier  date,  Sept.  21,  1886  =  1886  9  21 

4  9  21 
The  former  method  is  preferred  in  this  text. 

PROBLEMS. 

Note.     Tlie  teacher  should  name  the  method  for  each  problem. 

1.  "What  is  the  interest  on  $748.40,  at  7%,  from  June  12, 
1S85,  to  Dee.  21,  1888? 

2.  What  is  the  interest  on  $548.50,  at  8%,  from  July  18, 
1886,  to  May  5,  1890? 

3.  What  is  the  interest  on  $93.80,  at  5^%,  from  Dec.  7, 
1885,  to  Oct.  1,  1892? 

4.  Find  the  amount  of  $2480,  on  interest  at  6%,  from 
Aug.  28,  1884,  to  Feb.  19,  1889. 

5.  Find  the  amount  of  $3728.30,  on  interest  at  7%,  from 
Feb.  29,  1888,  to  Jan.  14,  1892. 

Find  the  interest  on  : 

6.  $469.12,  at  6%,  from  June  8,  1887,  to  Aug.  15,  1890. 

7.  $48.16,  at  7%,  from  Dec    15,  1885,  to  May  3,  1888. 

8.  $907.92,  at  77. ,  from  May  7,  1890,  to  Sept.  19,  1892. 

9.  $1359.06,  at  7%,  from  Oct.  24,  1888,  to  April  6,  1891. 

10.  $750.20,  at  5%,  from  March  12, 1886,  to  Nov.  28, 1886. 

11.  $609.47,  at  57  ,  from  Aug.  16,  1891,  to  June  1,  1892. 

12.  $1936.82,  at  41%,  from  Jan.  29,  1892,  to  Oct  14,1892. 


264  NEW  ADVANCED  ARITHMETIC. 

13.  $207.49,  at  4^%,  from  Nov.  3,  1883,  to  July  28,  1889. 

14.  ^2382.75,  at  8%,  from  Feb.  15,  1880,  to  Oct.  11,  1883. 

15.  S69.20,  at  8%,  from  April  1,  1888,  to  Jan.  22,  1893. 

16.  $7512.36,  at  8%,  from  July  30,  1888,  to  Dec.  9,  1888. 

17.  8169.17,  at7i%,from  Sept.  7,1885,  to  March  18, 1892. 

18.  S273.48,  at  71%,  fron  Dec.  19,  1886,  to  Nov.  30,  1890. 

19.  $428.10,  at  6% ,  irom  Feb.  23,  1887,  to  Aug.  5,  1891. 

20.  $491.73,  at  6%,  from  Nov.  16,  1886,  to  Nov.  28,  1886. 

21.  S636.80,  at  6% ,  from  Jan  5,  1889,  to  Jan.  31,  1893. 

Find  the  amount  of : 

22.  $19.12,  at  5.\%,  from  Aug.  12,  1883,  to  June  10,  1887. 

23.  $729.13,  at  51%,  from  June  18,  1886,  to  Oct.  21, 1890. 

24.  $258.18,  at  5^% ,  from  Dec.  3,  1890,  to  May  1,  1892. 

25.  $371.29,  at  5i%,  from  March  25, 1889,to  July  29,1891. 

26.  $580.00,  at  9%,  from  Jan.  16,  1885,  to  Sept.  10,  1889. 

27.  $412.31,  at  9%,  from  Oct.  2,  1888,  to  March  28,  1893. 

28.  $7539.06,  at  3i%,  from  Feb.  19, 1891,  to  Nov.  6, 1892. 

29.  $117.59,  at3%,  from  AprU  29, 1882,  to  Dec.  19,  1887. 

30.  $396.16,  at  6% ,  from  July  5,  1885,  to  Feb.  26,  1888. 

31.  $872.38,  at  7% ,  from  May  11,  1887,  to  Jan.  7,  1892. 

32.  $250.10,  at  7i%,  from  Nov.  17,  1884,  to  Aug.  24,  1890. 

33.  $1536.81,  at 6i%  ,  from  Sept.  23, 1889,  to  Jan.  17, 1893. 

34.  $2140.60,  at  6%,  from  June  6,  1890,  to  April  23,  1894. 

35.  $16.31,  at  8%,  from  Oct.  12,  1884,  to  March  5,  1889. 

36.  $589.76,  at  7% ,  from  Feb.  8,  1888,  to  Aug.  19,  1893. 

37.  $26824,  at  4%,  from  Dec.  18,  1890,  to  Feb.  21,  1894. 

297.     The  Method  of  finding  Interest  by  Days. 

1.  Interest  is  sometimes  computed  by  days,  when  the 
time  is  short.  In  such  cases  find  the  actual  number  of  days 
by  the  following : 


APPLICATIONS   OF  PERCENTAGE. 


265 


Method. 
July  15 : 


The  actual  number  of  days  from  March  10  to 


March,  21  days 

AprU,    30  " 

May,     31  " 

June,     30  " 

July,  _25  " 

127  " 

method  is  most  convenient  for  this 


6% 


2.  The  second 
purpose. 

3.  Since  the  interest  on  Si  for  6  days  at  6%  is  1  mill,  if 
the  number  of  days  be  divided  by  6,  the  result  will  be  the 
interest  in  mills  on  $1  at  6%.  To  find  the  interest  at  any 
other  rate,  find  such  a  part  of  this  result  as  the  given  rate 
is  of  6%. 

Illustrative  Problem.  Find  the  interest  on  $850,  at  5%, 
from  April  9  to  July  12,  same  year. 

TIME.  6850.00 


21 
31 
30 
12 

6)94 

15§  mills  =  interest  on 
at  6%. 


.015? 


4.250 

8.500 

.283 

.283 

$13,316 

i,  2.219 

$11,097 


298. 


1. 
2. 
3. 
4. 


Principal. 

$640.80 

$1254.25 

$796.13 

$2958 


PROBLEMS. 

Time. 

March  15,  1891,  to  June  24,  1891 
June  24,  1891,  to  Sept.  18,  1891 
Aug.  12,  1891,  to  Dec.  27,  1891 
May     17,  1891,  to  Aug.  29,  1891 


Rafe. 

6% 
7% 
5% 


$4872.80       Sept.    28,  1891,  to  Jan.      5,  1892      8% 


266  NEW  ADVANCED  ARITHMETIC. 

Principal.  Time.  Rate. 

6.  $86.97  Jan.        5,  1891,  to  June   16,  1891      9% 

7.  $916.20         Oct.      12,  1891,  to  Feb.      2,  1892    10% 

8.  88712.91       Dec.     27,  1891,  to  April  30,  1892      1\% 

299.  ACCURATE   INTEREST. 

Since  the  common  year  contains  365  days,  in  counting 
the  year  as  360  days,  each  day  is  counted  at  |-|  of  its  real 
length  as  compared  with  a  year.  Hence  the  interest  found 
by  the  360-day  method  is  \%  of  the  "accurate "  interest. 
To  find  the  interest  for  a  given  number  of  days  by  the  365- 
day  method,  subtract  ^^  of  the  result  obtained  by  the  360-day 
method,  and  the  difference  will  be  the  "  accurate  interest  by 
days." 

Find  the  accurate  interest  on ; 

1.  8824.50  for  120  days  at  6%. 

2.  $695.30  for  90  days  at  7%. 

3.  $1275  for  180  days  at  8%. 

4.  $2560  for  66  days  at  5%. 

5.  $88.40  for  48  days  at  6%. 

6.  $360  for  200  days  at  7%. 

7.  $940  for  300  days  at  6%. 

8.  $4875  for  96  days  at  7%. 

300.  PARTIAL   PAYMENTS. 

$800.  Boston,  Mass.,  Jan.  8,  1892. 

Six  months  after  date,  for  value  received,  I  promise  to 
pay  to  James  B.  Rogers,  or  order.  Eight  Hundred  Dollars, 
with  interest  at  seven  per  cent  per  annum. 

John  T.  Walker. 

1.  The  above-written  promise  is  a  Promissory  Note. 
Walker  is  the  Maker,  and  Rogers  the  Payee.  The  $800 
is  the  Principal. 


APPLICATIONS   OF  PERCENTAGE.  267 

2.  From  the  foregoing,  form  a  definition  of  a  promissory 
note,  and  of  maker  and  payee. 

3.  Promissory  notes  may  be  bought  and  sold  like  other 
forms  of  property.  Some  calculation  is  usually  necessary 
to  determine  the  value  of  a  note.     Why? 

4.  If  James  B.  Rogers  should  sell  the  above  note  he 
would  write  his  name  on  the  back.  This  is  called  "in- 
dorsing the  note  in  blank."  It  is  an  order  to  John  T. 
Walker  to  pay  it  to  the  person  who  owns  it  at  its  maturity, 
July  8,  1892.  The  indorsement  might  indicate  some  per- 
son upon  whose  order  the  note  is  to  be  paid.  This  is  a 
"special  indorsement." 

5.  Under  the  laws  of  some  States  the  above  indorse- 
ments would  make  Rogers  responsible  for  the  payment  of 
this  note  in  case  Walker  should  fail  to  do  so.  If  he  wishes 
to  avoid  such  responsibility  he  adds  the  words  "  without 
recourse  "  after  his  name. 

6.  There  are  many  forms  of  notes,  but  the  differences 
are  slight.  If,  instead  of  "  six  months  after  date,"  the 
words  "  on  demand  "  occurred  in  the  above  note  it  would 
be  called  a  Demand  Note.     Write  one. 

7.  If,  instead  of  "  I  promise,"  it  read,  "we  or  either  of 
us  promise,"  and  two  persons  signed  it,  it  would  be  a  Joint- 
and-Several  Note. 

8.  The  place  at  which  a  note  is  to  be  paid  is  often  indi- 
cated in  the  note.  It  will  usually  be  a  bank.  In  some 
States,  notes,  payable  at  banks,  are  not  legally  due  until 
three  days  after  their  apparent  maturity.  Such  time  is 
called  Days  of  Grace.  Interest  is  charged  for  them  if  the 
note  bears  interest. 

Note.  If  a  note  is  payable  to  "  bearer  "  it  may  be  exchanged  without 
indorsement.  If  made  payable  to  a  particular  person,  and  does  not  con- 
tain the  words  "  or  order"  or  "  or  bearer,"  it  is  not  negotiable. 

9.  Partial  payments  are  sometimes  made  on  notes  and 
other  evidences   of   indebtedness  that  bear  interest.     The 


268  NEW  ADVANCED  ARITHMETIC. 

United  States   Supreme   Court   has   prescribed   a   rule  foi 
findiug  their  value  at  any  time.     It  is  called 

301.    THE   UNITED   STATES    RULE. 

I.  The  rule  for  casting  interest,  tchen  vartial  pay- 
ments  have  been  made,  is  to  apply  the  paytnent,  in  the 
first  place,  to  the  discharge  of  the  interest  then  due. 

II.  If  the  payment  exceeds  the  interest  the  surplus 
goes  toicards  discharging  the  principal,  and  the  subse- 
quent interest  is  to  be  computed  on  the  balatice  of  prin- 
cipal remaining  due. 

III.  If  the  payment  be  less  than  the  interest  the  sur- 
plus  of  interest  must  not  be  taken  to  augment  the 
principal ;  but  interest  continues  on  the  former  principal 
until  the  period  tvhen  the  payments,  taken  together,  ex- 
ceed the  interest  due,  and  then  the  surplus  is  to  be  ajtplied 
towards  discharging  the  principal ;  and  interest  is  to  be 
eotnputed  on  the  balance  as  aforesaid* 

Illustrative  Problem.  A  note  of  $850,  bearing  interest 
at  8%,  and  dated  March  1,  1885,  has  the  following  in- 
dorsements : 

Jan.  24,  1886.        $86. 25. 

March  12,  1887.     $72.00. 

Jan.  5,  1888.        $153.50. 

Feb.  20,  1889.     $265.80. 
What  was  due  March  1,  1890? 

Form. 

March    1,  1885 
$86.25        Jan.      24,  1886 1 
S72.00         March  12,  1887  H  3^- 
$153.50        Jan.        5,  1888 1 
$265.80        Feb.     20,  1889  {  J  |^ 
March    1,  1890  >  ^  " 

Principal  $850 

m 

$08.00      =  interest  for    1  yr. 


10 

mo. 

23  d. 

1 

(( 

16  « 

9 

« 

24  « 

1 

« 

15" 
9" 

APPLICATIONS   OF  PERCENTAGE. 


269 


6  mo.  =  4  of  1  yr. 
;          4   "     =  \  of  1  " 
20  d.     =  ^  of  4  mo. 

2  "      =   L  of  20  d. 
1  "      =  i  of  2  d. 

1  mo.  =  Y*5  of  1  yr. 
10  d.     =  J  of  1  mo. 
6  "      =  1  of  1    " 

Payment 

6  mo.  =  4  of  1  yr. 

3  "     =  ^  of  6  mo. 
15  d.     =iof3   " 

9  "      =  J^  of  3  " 

2d  payment  S72.00 
3d        «'      S153.50 

1  mo.  =  J5  of  1  yr. 
15  d.     =  ^  of  1  mo. 

$34.00 

22.666 

3.777 

.377 

.188 

=  interest  for  6  mo. 
=        "        "4   " 
"   20  d. 
_        u        "2  " 
-_        «        "1  " 

$61,008 
850 

=        "        "    10  mo.  23  d. 
=  first  principal. 

$911,008 
86.25 

=  amount,  Jan.  24,  1886. 

=  first  payment. 

$824,758 
.08 

=  new  principal,  Jan.  24,  1886 

$65.9800 
5.498 
1.832 
1.099 

$74,409 

$72.00 

=  interest  for    1  yr. 
=        "        "1  mo. 
=        "        "    10  d. 
=        "        "6  " 

=        "         "      1  yr.  1  mo.  16  d. 

less  than  interest. 

$65.98 

=  interest  for    1  yr. 

$32.99 
16.495 
2.749 
1.649 

=       "        "6  mo. 

—          a           «       3    (( 

"    15  d. 

:=         "          "9  " 

$53,883 

74.409 

824.758 

=        "        "9  mo.  24  d. 
=  previous  interest. 
=  second  principal. 

$953,050 
225.50 

=  amount,  Jan.  5,  1888. 
2d  and  3d  payments. 

$727.55 
.08 

=  3d  principal. 

$58.2040 
4.8503 
2.4251 

=  interest  for    1  yr. 
=        "         "1  mo. 
=        "         "    15  d. 

$65.4794 
727.55 

=        "        "      1  yr.  1  mo.  15  d. 

270  NEW  ADVANCED  ARITHMETIC. 

$793,029    =  amount,  Feb.  20,  1889. 
265.80      =  4th  payment. 

$527,229    =  4th  principal. 
.08 

S42.1776  =  interest  for  1  yr. 

$3,514  =  interest  for  1  mo. 
6  d.      =  ^  of  1  mo.  .703    =  interest  for  6  d. 

3  "      =  ^  of  6  d.  .351    r=        "        "3  " 

$43,231    =        "         "    1  yr.  9  d. 
527.229    =  4th  principal. 


$570,460    =  amount  due  March  1,  1890. 
Note.     Payments  less  than  the  accumulated  interest  will  rarely  be 
made,  since  they  do  not  diminish  the  interest-bearing  portion  of  the  debt. 

1.  A  note  whose  principal  is  $500,  dated  March  1,  1890, 
and  bearing  interest  at  6%,  has  the  following  indorsements: 

June  1,  1891,  $65. 

Sept.  16,  1892,     $124. 
What  was  due  Jan.  1,  1894? 

2.  Principal,   $850.     Date,   May   10,    1891.     Rate,    7%. 
Indorsements : 

July  15,  1892,     $130. 
June  1,  1893,        $46. 
Dec.  12,  1894,     $380. 
What  was  due  May  10,  1895? 

3.  Principal,  $1,000.     Date,  Sept.   1,   1892.     Rate,  8%. 
Indorsements : 

March  12,  1893,       $75. 
June  18,  1894,        $275. 
March  15,  1895,     $360. 
What  was  due  Sept.  1,  1895? 

4.  Principal,  $1,200.     Date,  July  1,    1890.     Rate  5i%. 
Indorsements : 

^^  March  16,  1891,  $160. 

June  12,  1892,  $320. 

Aug.  5,  1893,  $500. 
What  was  due  July  1,  1894? 


APPLICATIONS   OF  PERCENTAGE.  271 

5.  A  note  of  $1,800,  bearing  interest  at  7%,  and  dated 
June  12,  1886,  was  indorsed  as  follows : 

March  21,  1887,     S183.50. 
Oct.  12,  1888,        $395.75. 
May  10,  1890,        S583.45. 
What  was  due  July  1,  1892? 

6.  $1,250.  Bloomington,  III.,  Oct.  15,  1886. 
Four  years  after  date,  for  value  received,  we,  or  either  of 

us,  promise  to  pay  to  W.  O.  Davis  &  Co.,  or  order.  Twelve 
Huntired  Fifty  Dollars,  with  interest  at  6%  per  annum. 

James  T.  Ronet, 
John  J.  Condon. 
The  following  statements  were  written  across  the  back  of 
this  note : 

"  Bec'd  on  the  within  note,  Dec.  1,  1887,  $358.80." 
"  Rec'd  on  the  within  note,  Jan.  21,  1889,  $475.00." 
"  Rec'd  on  the  within  note,  Oct.  25,  1889,  $261.50." 
"  Rec'd  on  the  within  note,  June  15,  1890,  $91.40." 
What  was  due  Oct.  15,  1890? 

7.    A  note  for  $2,580,  dated  July  12,  1884,  bearing  5% 
interest,  had  the  following  indorsements  : 
Jan.  1,  1885,  $75. 

May  25,  1885,  $87.40. 

Dec.  18,  1885,         $260. 
Oct.  15,  1886,         $326.45. 
June  28,  1887,         $752.31. 
Nov.  12,  1888,         $850. 
What  was  due  July  12,  1889? 

8.    Principal,  $762.84.     Date,  Sept.   24,   1886.     Rate  of 
interest,  10%,     Indorsements: 

Jan.  1,  1887,  $51.80. 

Aug.  23,  1887,        $128. 
May  17,  1888,        $125. 
Oct.  28,  1889,         $214.80. 
March  13,  1890,     $306.90. 
What  was  due  Sept.  24,  1890? 


272  NEW  ADVANCED  ARITHMETIC. 


9.    Principal,  $8,750.     Rate, 

5%.     Date,  April  12,  1882. 

Indorsements  : 

June  20,  1883, 

$1,250. 

Aug.  3,  1884, 

$2,560. 

Dec.  25,  1885, 

$3,164.86. 

July  30,  1886, 

$1,571.29. 

Dec.  15,  1887, 

$1,262.80. 

What  was  due  Oct.  12,  1888? 

10.    Principal,  $2,350.     Date, 

Aug.  3,  1885. 

Rate,  71%. 

Indorsements  : 

Sept.  5,  1886, 

$250. 

Jan.  1,  1888, 

$60. 

July  25,  1888, 

$475. 

March  15,  1889,     $560. 

Aug.  3,  1890, 

$880. 

What  was  stiU  due? 

11.   Principal,  $3,000.     Date, 

Jan.  10,  1886. 

Rate,  6%. 

Indorsements : 

March  1,  1887 

$260. 

June  11,  1888, 

$624. 

Aug.  25,  1889, 

$1,030. 

May  1,  1891, 

$1,250. 

Jan.  10,  1892, 

$280. 

What  was  still  due  ? 

302.      THE    MERCHANTS'    RULE. 

I.  Find  the  amount  of  the  principal  for  the  entire  tintCt 

II.  Find,   the  amount  of  each  payment  from  the  time 
that  it  tvas  made  to  the  time  of  settlement. 

III.  From   the    first   amount   subtract   the   sum   of  the 
amounts  of  the  serei'al  payments. 

Note.    This  method  allows  interest  on  each  payment  for  all  of  the  time 
that  it  is  in  the  creditor's  possession.    It  is  a  perfectly  fair  method. 

1.   A  note  for  $650,  dated  Jan.  10,  1894,  and  bearing  in- 
terest ai  5%,  has  the  following  indorsements:  March   15, 


APPLICATIONS   OF  PERCENTAGE.  273 

$125;  July  12,  S240 ;  Oct.  5,  $85.     What  was  due  Jan.  10, 
1895  ? 

2.  A  note  for  $785.40,  dated  May  1,  1895,  and  bearing 
interest  at  6%,  is  indorsed  as  follows:  Aug.  20,  $180.20; 
Nov.  5,  $250.80;  Feb..  24,  1896,  $236.50.  What  was  due 
May  1,  1896? 

3.  Principal,  $892.60.  Date,  July  25,  1895.  Rate,  7%. 
Indorsements:  Sept.  1,  $325;  Nov.  19,  $175.50;  Jan.  12, 
1896,  $90;  May  10,  $300.     What  was  due  July  1? 

4.  Principal,  $1,280.  Rate,  8%.  Date,  May  1,  1894. 
Indorsements:  July  25,  $300;  Sept.  10,  $250;  Dec.  1,  $350; 
Feb.  10,  1895,  $100.     What  was  due  April  20,  1895? 


303.    ANNUAL   INTEREST. 

1.  Annual  interest  differs  from  compound  interest  in  one 
particular :  interest  does  not  draw  compound  interest,  but 
simple  interest. 

A  problem  will  make  the  difference  clear. 

2.  If  a  note  provides  that  interest  is  payable  annually,  it 
means  that  unpaid  interest  at  the  end  of  any  period  shaU 
draw  simple  interest  until  paid. 

niustrative  Example.  A  note  of  $400  is  due  in  4  yr. 
6  mo.  The  interest  at  7%  is  payable  annually.  If  nothing 
is  paid  until  the  note  is  due,  what  will  the  interest  amount 
to?  The  interest  on  $400  for  4  yr.  6  mo.  =  $126.  The 
$28  due  at  the  end  of  the  first  year  draws  interest  for  3J 
years ;  the  $28  due  the  second  year,  for  2i  j-ears ;  the  third, 
for  1^  years  ;  and  the  fourth  for  h  year.  There  will  then  be 
due,  in  addition  to  the  $126,  the  interest  on  $28  for  3i  yr. 
+  2^  yr.  +  li  yr.  +  ^  yi'.  =  the  interest  for  8  years  = 
$15.68.     $126  +  $15.68"=  $141.68. 

Make  a  rule  from  the  above  analysis. 


NEW  ADVANCED  ARITHMETIC, 


PROBLEMS. 

Principal. 

Time 

RatA 

1. 

$350 

3 

yr- 

5  mo.    6  d.            6% 

2. 

$400 

4 

'^ 

7    ' 

'      9  ' 

'             6% 

3. 

8480 

2 

8    ' 

'    12  ' 

7% 

4. 

$510.40 

4 

10    ' 

'     15  ' 

7% 

5. 

$560 

5 

6    ' 

'    18  ' 

7% 

6. 

$85.50 

3 

11    ' 

'    19  ' 

7% 

7. 

$128.20 

4 

2    ' 

'    21  ' 

8% 

8. 

$649 

3 

4    ' 

'    24  ' 

8% 

9. 

$763.50 

6 

7    ' 

'    20  ' 

5% 

10. 

$840 

5 

1    ' 

'       1  ' 

5% 

11. 

$24.10 

7 

7    ' 

'      7  ' 

5i% 

12. 

$968 

6 

6    ' 

'       6  » 

6% 

13. 

$1070 

4 

2    ' 

6% 

14 

$312.40 

5 

6    ' 

'    20  ' 

6% 

15. 

$2060 

2 

9     ' 

'     10  ' 

6% 

16. 

$3840 

3 

4 

6% 

17. 

$5625 

4 

9     ' 

'     12  ' 

7% 

18 

$96.90 

5 

8    ' 

'     15  ' 

4i% 

19. 

$4500 

6 

3     ' 

'    17  ' 

41% 

20. 

$6970 

4 

5     ' 

'      5  ' 

5% 

21. 

$386.75 

5 

10    ' 

'    10  ' 

5% 

22. 

$193.40 

6 

6  ' 

5i% 

23. 

$4200 

4 

9  ' 

5J% 

24 

$10000 

3 

3    ' 

'      3  ' 

4i% 

25. 

$693.42 

4 

11     ' 

'      6  ' 

5% 

304.    COMPOUND   INTEREST. 

1.  A  man  borrowed  $350  at  7%  interest,  agreeing  that 
if  the  interest  were  not  paid  at  the  end  of  the  first  year  it 
should  be  added  to  the  principal  to  make  a  new  principal  for 


APPLICATIONS   OF  PERCENTAGE.  275 

the  second  year,  and  that  the  interest  should  be  added  thus 
each  year  until  the  debt  was  paid.  If  he  paid  nothing  until 
the  end  of  the  3  years  and  3  months,  how  much  was  then 
due? 

1st  principal  $350 

1st  year's  interest  24.50 

2d  principal  $374.50 

2d  year's  interest  26.215 

3d  principal  $400,715 

3d  year's  interest  28.049 

4th  principal  $428,764 

Interest  for  3  mo.  7.502 


Amount  $436,266 

1st  principal  350 

Interest  $86.26 

2.  Compound  interest  is  interest  upon  a  principal  that  is 
increased  at  regular  periods  by  its  accumulated  interest. 

3.  Interest  may  be  compounded  at  the  end  of  any  period 
agreed  upon,  instead  of  annually,  as  above. 

RULE 
To  calculate  compound  interest: 

1.  At  the  end  of  each  iteriod  increase  the  principal  for 
that  period  by  the  interest  accumulated  during  the  period. 

2.  From  the  final  amount  subtract  the  first  principal. 

305.     PROBLEMS. 

Find  the  compound  amount  and  interest : 

1.  Of  $528,  for  3  years,  at  5%. 

2.  Of  $1200,  for  4  years,  at  8%. 

3.  Of    $1680.50,    for    2    years,    at    10%,    compounding 
quarterly. 


276 


NEW  ADVANCED  ARITHMETIC. 


4.    Of  $2560,  for  3  yr.  8  mo.  25  d.,  at  6%,  compounding 
semi-annually. 

306.    TABLE, 

Showing  the  amount  of^\  at  compound  interest  from  1  year  to  10  years, 
at  3,  4,  4^,  5,  6,  ant/  7  per  cent. 


Years. 

1 

3  per  cent. 

4  per  cent. 

4J  per  cent. 

5  per  cent. 

G  per  cent. 

7  per  cent. 

1.030000 

1.040000 

1.045000 

1.0.50000 

1.060000 

1.070000 

9 

1.060900 

1.081600 

1.092025 

1.102500 

1.123600 

1.144900 

3 

1.092727 

1.124864 

1.141166 

1.1,57625 

1.191016 

1.22.5043 

4 

1.12,5.509 

1.169859 

1.192519 

1.215506 

1.262477 

1.310796 

5 

1.159274 

1.216653 

1.246182 

1.276282 

1.338226 

1.402552 

6 

1.194052 

1.265319 

1.302260 

1.340096 

1  418519 

1.500730 

7 

1.229874 

1.315932 

1.360862 

1.407100 

1.. 503630 

1.005781 

8 

1.266770 

1.368.569 

1.422101 

1.477455 

1.593848 

1.718186 

9 

1.304773 

1.42.3312 

1.486095 

1.551328 

1.689479 

1.8384,59 

10 

1.343916 

1.480244 

1.552969 

1.628895 

1.790848 

1.967151 

Illustrative  Example.     Find  compound  interest  of   $1UU 
for  8  yr.  4  mo.  12  d.,  at  5%. 


Amount  of  $1.00  for  8  yr.  at  5% 

$1.4774 

100 

Amount  of  .$100 

S147.74 

Interest  of  $147.74  for  4  mo.  12  d. 

2.70 

Amount 

$150.44 

100.00 

Compound  interest  $50.44 

With  the  aid  of  the  table  find  the  amount  and  compound 
interest : 

Principal.  Time.  Rate. 

1.  $750                     4  yr.  6  mo.  i\%. 

2.  $5,000                 9  "  7    "    10  d.  6%. 

3.  $1,275.45            8  "  2    "     18  "  7%. 


APPLICATIONS   OF  PERCENTAGE. 


277 


Principal. 

Time. 

Rate. 

4. 

$1,640.25 

4 

yr- 

5  mo.  24  C 

I.             5%. 

5. 

$2,850.66 

7 

1     ' 

10"              4%. 

6. 

$834 

5 

7    " 

3%. 

7. 

$796 

8 

5    ' 

10  ' 

4%. 

8. 

$1,028.50 

9 

2    " 

12  ' 

41%. 

9. 

$1,725.80 

3 

10    ' 

15  ' 

5%. 

10. 

$9G0 

4 

1    ' 

21  ' 

6%. 

11. 

$2,480 

7 

11    " 

24  ' 

7%. 

12. 

$3,812 

6 

3    ' 

28  ' 

6%. 

13. 

$86.95 

2 

7    ' 

'     19  ' 

5%. 

14. 

$731.25 

3 

4    ' 

17  ' 

41%. 

15 

$5,960 

4 

5    ' 

11  ' 

3%. 

16. 

$72.15 

5 

8    ' 

16  ' 

4%. 

17. 

$3,824 

6 

2    ' 

18  ' 

5%. 

18. 

$7,500 

7 

10    ' 

'     12  ' 

3%. 

19. 

$438.90 

9 

4    ' 

15  ' 

6%. 

20. 

$1,500 

10 

10    ' 

'    10  ' 

7%. 

21. 

$2,500 

4 

1    ' 

1  ' 

4i%. 

Principal. 

Time. 

Ra 

e. 

22. 

$800      4  yr 

.  4  mo. 

6^ 

6-  comf 

)ounded  semi- 

Note.     Use  kalf  the  rate  for  double  the  time. 

23.  $1,500  3yr.  6mo.  24d.     8%  compounded  semi-aun'ly. 

24.  $1,850  4  "   8    "    20  "      9%  "  " 

25.  $2,500  2   "   5    "    12  "  12%  "  quarterly. 
Note.     Use  one-fourth  the  rate  for  four  times  the  time. 

26.  $3,150  3jT.  7mo.  18d.  8%  compounded  semi-ann'ly. 

27.  $3,675  4   "  2    "      6  "  6%  "  " 

28.  $4,180  5   "  3   "    10  "  6%  "  " 

29.  $5,000  3   "  7   "    15  "  8%  "  " 

30.  $10,000  4  "10   "    18  "  6%  " 

lyA 


278 


NEW  ADVANCED  ARITHMETIC. 


307.     GENERAL   PROBLEMS   IN   SIMPLE    INTEREST. 

1.  "We  have  seen  that  five  elements  have  appeared  in 
problems  in  interest.  These  are  principal,  interest,  amount, 
rate  per  cent,  and  time.  The  relations  between  them  are 
such  that  if  any  three  of  them  be  given,  the  other  two  may 
be  found.  Several  different  problems,  consequently,  may 
arise. 

308.  Problem   I. 

Given  the  principal,  rate  per  cent,  and  time,  to  find  the 
interest. 

This  problem  has  been  discussed  sufficiently. 

309.  Problem   II. 

Given  the  principal,  interest,  and  time,  to  find  the  rate 
per  cent. 

Illustrative  Problem.  The  interest  on  $324.00  for  o  yr. 
6  mo.  15  d.  is  $91. 'J7.     What  is  the  rate  per  cent? 

AxALYSis.  The  interest  on  S324.60  for  3  yr.  6  rno.  15  d.,  at  1%.  is 
$U.49|.  To  produce  $91.97  in  the  same  time,  the  rate  must  be  as  maiiv 
times  1%  as  $91.97  is  times  $11.49f.  $91.97  is  8  times  $ll.49|;  hence,  tlio 
required  rate  most  be  8%. 

PROBLEiVIS. 
Principal,  Interest.  Time. 

$66.13        Syr.    5  mo.  10  d.    Find  rate. 
$85.36        2 


1. 
2. 
3. 
4. 
5. 
6. 
7. 
8 
9. 
10. 


$650.40 
$864.36 
$1,275.86 
$1,464.29 


$242.89  4 
$180.04  1 
$306.28  2 
$2,580.47  $577.02  3 
$3,600  $951.75  5 
$4,,580  $1,134.94  4 
$5,000  $1,6 63.. 38  5 
SO. 840. 75  $3,575.72  6 


7 
8 
9 
1 
2 

10 
6 
3 

11 


15 
6 
5 
3 

10 

15 
2 
1 

19 


APPLICATIONS    OF  PERCENTAGE. 


279 


310.     Problem    III. 

Given  the  principal,  interest,  and  rate  per  cent,  to  find 
the   time. 

Illustrative  Problem.  The  interest  on  $324.60,  at  8%, 
was  $91.97.     What  was  the  time? 

Analysis.  The  interest  on  $324.60  for  1  month,  at  8%,  is  $2,164.  To 
produce  $91.97  at  the  same  rate,  the  time  must  be  as  many  times  1  month 
as  $91.97  is  times  $2,164.  $91.97  is  42i  times  $2,164;  hence,  the  time 
was  42|  months,  which  =  3  yr.  6  mo.  15  d. 


Priuiipal. 

Interest. 

Rate. 

1. 

$490.92 

849.75 

8%. 

Find  time 

2. 

83,794.08 

8803.40 

9%. 

3. 

S892.-15 

8149.73 

5%. 

4. 

81,231.16 

81G9.08 

6%. 

5. 

$0,871.48 

81,701.10 

7%. 

6 

S89.26 

818.04 

3%. 

7. 

813.51 

84.37 

5i-%. 

8. 

81.05 

80.55 

6,\%. 

9. 

810,874.80 

8193.33 

4%. 

10. 

8916.15 

858.63 

8%. 

311.     Problem    IV. 

Givsn  the  interest,  rate  per  cent,  and  time,  to  find  the 
principal. 

Illustrative  Problem.  What  principal  will  produce  891.97 
in  3  yr.  6  mo.  15  d.,  at  8%  ? 

Analysis.  A  principal  of  $1.00,  with  the  above  rate  and  time,  will 
produce  $0.28j.  To  produce  $91.97,  the  principal  must  he  as  many  times 
$1.00  as  $91.97  is  times  $0.28i;  $91.97  is  324.6  times  $0.28^;  hence,  the 
required  principal  is  $324.60. 

Note.     Change  divisor  and  dividend  to  thirds. 


280 


NEW  ADVANCED  ARITHMETIC. 


Interest. 

Rate. 

Time. 

1. 

S184.67 

7% 

3 

yr.  6  mo.    9  d. 

2. 

$166.62 

8% 

3 

"   9    ' 

'     17  " 

3 

$35.16 

51% 

6 

"    9    ' 

'    24  " 

4. 

$89.68 

6% 

3 

u    2    ' 

'      7  " 

5. 

$8.04 

7% 

2 

tt    4    t 

'     18  " 

6. 

$150.41 

7% 

2 

4t      4.       4 

'    12  " 

7. 

$233.34 

7% 

3 

"    5    ' 

'    13  " 

8. 

$26.67 

5% 

9    ' 

'    15  " 

9. 

$24.21 

5% 

8    ' 

'     15  " 

10. 

$61.74 

^% 

5 

"   8    ' 

-    25  " 

Find  principaL 


312.     Problem  V. 

Given  the  amount,  rate  per  cent,  and  time,  to  find  the 
principal. 

Illustrative  Prohlera.  What  principal  will  amount  to 
$416.57  in  3  yr.  6  mo.   15  d.  at8%? 

Analysis.  A  principal  of  $1.00  will  amount  to  $1.28^  with  the  alx)ve 
time  and  rate.  To  amount  to  $416..57,  the  principal  must  be  as  many 
times  $1.00  as  $416..57  is  times  $1.28^.  $416.57  is  324.6  times  $1  28^; 
hence,  the  reciuired  principal  is  $324.60. 

Note.  Observe  that  we  first  found,  in  Art.  309,  what  a  1  %  rate  would 
produce;  in  Art.  310,  what  would  be  produced  in  1  month;  in  Art.  311, 
what  a  $1.00  principal  would  yield  ;  and  in  Art.  311,  what  a  $1.00  prin- 
cipal would  amount  to.  From  these  illustrative  problems  the  following 
general  statement  may  be  derived  :  Perform  the  problem,  assuming  one  of 
the  required  kind  to  he  the  required  answer  ;  then  compare  the  result  with 
the  given  numt)er  of  the  same  kind. 


PROBLEMS. 
Amount.  Rafe.  Time. 

1.  $460  5%       2  yr.    3  mo.  20  d.  Find  principaL 

2.  $380.80  6%       3  "      4    "     15  "        "  " 

3.  .^524. 10  7%        4  "      6    "     12  "         "  " 

4.  $736,50  8%        1  "      9    "     16  "         "  " 


APPLICATIONS   OF  PERCENTAGE. 


281 


Amount. 

Rate. 

Time. 

5. 

$6'Ja.l2 

7% 

5  mo. 

24  d. 

Find  principal 

6. 

$874 

H% 

Syr. 

11     " 

19  " 

H                     4  t      ' 

7. 

81)26.95 

41% 

5  " 

2    " 

21  " 

8. 

$1,284 

51% 

2  " 

7    " 

6  " 

9. 

$2,065.48 

7% 

3  " 

8    '> 

28  " 

10. 

$3,129.76 

6% 

1  " 

1    " 

18  " 

313.       PROBLEMS. 

1.  At  what  rate  will  $240  gain  $8.96  in  6  mouths  and  12 
days  ? 

2.  In  what  time  will  $145.60  gain  $42.47  at  5%  ? 

3.  What  principal  will  gain  $52.57  in  2  yr.  4  mo.,  at  6%  ? 

4.  What  principal  will  amount  to  $132.40  in  5  yr.  7  mo. 
20  d.,  at  10%. 

5.  At  what  rate  per  cent  will  any  principal  double  itself  in 
10  years?  8  years?  12  years?  16§  years?  20  years?  At 
what  rate  will  any  principal  treble  itself  in  the  same  periods  ? 

6.  In  what  time  will  a  principal  double  itself  at  3%  ?  4%  ? 

4^%  ?  5%  ?  51%  ?  6%  ?  7%  ?  8%  ?  9%  ?  10%  ?     In  what  time 
wUl  it  treble  itself  at  same  rates  per  cent? 

7.  What  was  a  man's  investment  that  yielded  him  an  in- 
come of  $1,264.75  in  1  yr.  7  mo.  24  d.,  at  7%  interest? 

8.  What  principal  will  amount  to  $5,860.80  in  3  yr.  10  mo. 
20  d.,  at  8%  interest? 

9.  At  what  price  must  bonds  that  bear  4%  interest  annu- 
ally be  bought,  to  yield  6%  interest  on  the  investment? 

10.  Bought  4%  bonds  at  70.  What  rate  per  cent  of  in- 
terest does  the  investment  yield  ? 

11.  Bought  a  piece  of  land  for  $6,825^  Kept  it  2  yr.  6 
mo.  18  d.,  and  sold  it  for  $9,841.65.  What  is  the  rate  per 
cent  of  interest  that  the  investment  yielded  ? 


282  NEW  ADVANCED  ARITHMETIC. 

12.  How  long  must  $586.40  be  on  interest  at  8%  to  gain 
$134.50? 

13.  AVhat  sum  of  money  must  be  invested  at  6%  interest, 
compounded  annually,  to  amount  to  SlO,000  in  10  years' 

14.  What  investment  will  yield  annually  §564.75  at  6%  ? 

15.  In  what  time  will  82,500  amount  to  S3,650  at  U'/c  ? 

16.  What  sum  put  at  interest,  Jan.  1,  1886,  will  amount 
to  §343.75,  Feb.  1,  1888,  at  7%  ? 

Find  the  lacking  numbers  in  the  following  problems : 


1. 

Principal. 

S834.60 

Rate. 

7% 

Time. 

9 

Interest. 

848.20 

Amount. 

2. 

6560.40 

? 

2  yr.  2  mo. 

854.639 

? 

3. 

p 

6% 

7  mo.  15 

d. 

? 

8350.65 

4. 

82,500 

8% 

? 

8556.67 

? 

5. 

V 

9% 

90  d. 

872.55 

? 

6. 

8150.60 

? 

72  d. 

82.26 

7. 

8425.75 

6% 

? 

88.52 

8. 

? 

8  7o 

93  d. 

812.60 

9. 

v 

7% 

33  d. 

8524.12 

10. 

81,530.38 

H% 

? 

8375.00 

314.     PRESENT    WORTH   AND    TRUE    DISCOUNT. 

1.  .Tames  K.  Briggs  purchased  from  Roliert  R.  Stone  a 
horse  and  carriage  for  8250,  with  the  understanding  tliat  ho 
was  to  pay  for  them  18  months  afterward,  without  interest. 
He  made  the  following  note  : 

8250.  RocHESTKR,  N.  Y.,  May  1,  1890. 

Eighteen  months  after  date,  for  value  received,  I  promise 
to  pay  to  Robert  R.  Stone,  or  order.  Two  Hundred  and  Fifty 
Dollars,  without  interest  until  due. 

James  K.  Briggs. 


APPLICATIONS   OF  PERCENTAGE,  283 

Ou  the  1st  day  of  July,  1890,  Stone  offered  this  note  for 
sale.  What  should  a  person  pay  for  it  so  that  at  its  matu- 
rity he  should  receive  his  purchase  money  and  interest  on  it 
at  6  %  per  annum  for  1 6  months  ? 

He  will  receive  $250  at  the  maturity  of  the  note.  The 
question  is,  What  principal,  at  6%  interest  per  annum,  will 
amount  to  $250?  A  principal  of  $1.00  will  amount  to  $1.08 
in  16  months  at  6%  per  annum.  To  amount  to  $250,  the 
principal  must  be  as  many  times  $1.00  as  $250  is  times 
$1.08.  $250  is  231.48+  times  $1.08;  hence,  the  required 
amount  is  $231.48+. 

The  problem  may  be  proved  by  finding  the  interest  on 
$231.48  for  the  given  time  at  6%. 

2.  The  Present  "Worth  of  a  debt  due  at  a  future  time,  and 
not  bearing  interest,  is  that  sum  which,  put  at  interest,  at  the 
given  rate,  for  the  given  time,  will  amount  to  the  debt. 

3.  The  True  Discount  of  a  debt  is  the  difference  between 
the  debt  and  its  present  worth. 

RULE. 

To  find  the  present  ivorth  of  a  debt,  divide  the  amount 
of  the  debt  at  its  maturity  by  the  amount  of  $1.00  for  the 
given  time  at  the  given  rate  of  interest. 

To  find  the  true  discount,  subtract  the  present  toorth 
from  the  debt. 

Note.  Observe  that  true  discount  is  the  interest  on  the  present  worth 
for  the  given  time. 

315.  Another  Method,  i.  Any  principal  at  6%  will 
amount  to  108%  of  itself  in  16  months.  In  this  case  the 
amount  is  $250.  $250,  therefore,  is  108%  of  the  present 
worth.  1%  of  the  present  worth  is  ^i^  of  $250.  100%  of 
the  present  worth  is  jg§  of  $250. 

Which  of  the  general  problems  of  percentage  is  this? 

2.  Observe  that  by  this  method  the  discount  may  be 
fonnd  without  finding  the  present  worth. 


284  NEW  ADVANCED  ARITHMETIC. 

The  discount  is  8%  of  the  present  worth.  We  have  seea 
that  1  %  of  the  present  worth  is  y  J  ^  of  the  amount ;  hence, 
8%  of  the  present  worth  is  yg^  of  the  amount. 

PROBLEMS. 

1.  "What  is  the  present  worth  of  a  non-interest-bearing 
debt  of  $1,250,  due  in  2  yr.  3  mo.  15  d.,  if  money  is  worth 
6%  ?     What  is  the  true  discount? 

2.  Principal,  $875.40;  time,  1  yr.  4  mo.  20  d. ;  rate,  7%. 
Find  present  worth  and  discount,  the  principal  not  bearing 
interest. 

In  the  following  problems  the  principal  does  not  bear 
interest.     Find  present  worth  and  discount. 


Principal. 

Time. 

Rate 

3. 

$580.30 

3yr. 

4  mo.  10  d. 

6% 

4. 

$671.95 

5    " 

7    a       8  u 

7% 

5. 

$2,834.25 

2    " 

5    "     12  " 

5% 

6. 

$10,000 

1    " 

3    "       6  " 

4% 

7. 

$383.59 

2    " 

11    "     21  " 

8% 

8. 

$789.13 

2    " 

9    "     26  " 

9% 

9.  What  is  the  present  worth  of  an  interest-bearing  debt, 
if  the  rate  of  discount  is  the  same  as  the  rate  of  interest? 

10.  A  note  dated  May  12,  1889,  for  $1,060,  due  Sept.  21, 
1891,  and  bearing  interest  at  5%,  was  discounted  Oct.  15, 
1890,  at  6%  .     What  was  its  worth? 

Note.  Remember  that  the  amount  due  at  the  maturity  of  the  uote  is 
the  debt  whose  present  worth  is  to  be  found. 

Find  present  worth  and  true  discount  of  the  following 
non-intei'est-bearing  notes. 

Principal.  Date.  Maturity.  D^fcount 

11.  $84.90        Jan.  1,  1890  March  12,  1892       5% 

12.  $120  June  12,  1892        Oct.  5,  1894  5% 

13.  $250.80      March  10,  1888     May  1,  1891  6% 


APPLICATIONS   OF  PERCENTAGE.  285 

Principal.  Date.  Maturity.  ^^^^[^ 

V  S360  Oct.  15,  1893  June  28,  1896  6% 

i.5.  S480.40  Feb.  20,  1892  Aug.  1,  1895  6|% 

16.  $560.70  July  28,  1890  Dec.  10,  1894  ^% 

17.  $1,290  Aug.  3,  1894  May  15,  1897  7% 

18.  $2,580  Nov.  19,  1893  Oct.  1,  1897  1\% 

19.  $1,624  Dec.  30,  1895  July  24,  1898  7^% 

20.  $3,560  April  20,  1896  Dec.  1,  1898  8% 

21.  $787.50  May  1,  1896  Jan.  19,  1899  S\% 

22.  $48  Sept.  21,  1896  March  5,  1899         9% 

23.  $6,000  Jan.  23,  1896  April  21,  1899         9% 

24.  $980  Dec.  15,  1895  June  12,  1898  10% 

25.  $25  Oct.  19,  1896  Aug.  1,  1898  10% 

316.    BANK   DISCOUNT. 

1.  If  the  note  in  Art.  314  had  been  sold  to  a  bank,  its 
value  would  have  been  estimated  somewhat  diflferently  from 
the  method  there  given.  The  banker  would  have  calculated 
the  interest  on  $250  for  16  months  and  3  days,  if  days  of 
grace  are  counted,  and  would  have  subtracted  the  result 
from  $250.  The  remainder  would  have  been  the  proceeds, 
or  avails,  or  cash  value,  of  the  note. 

2.  The  Bank  Discount  of  a  note  not  bearing  interest  is 
the  interest  upon  the  face  of  the  note  for  the  time  from  the 
day  of  discount  until  its  legal  maturity. 

3.  The  three  days  that  are  sometimes  added  to  the  time 
which  a  note  is  to  run  are  called  Days  of  Grace. 

4.  Several  of  the  States  have  done  away  with  days  of 
grace.  In  the  following  problems  take  no  account  of  them 
unless  they  are  mentioned. 


286  NEW  ADVANCED  ARITHMETIC. 

PROBLEMS. 
What  are  the  bank  discount  and  avails  of  the  following 
note,  if  discounted  at  6%  on  the  day  that  it  was  made? 

$600.  Chicago,  III.,  May  12,  1891. 

Ninety  days  after  date,  for  value  received,  I  promise  to 
pay  to  the  First  National  Bank  of  Chicago  Six  Hundred 
Dollars,  with  interest  at  6%  per  annum,  after  due. 

Benjamin  E.  Brown. 

Analysis.  The  interest  on  $600  for  93  days  at  6%  is  9.30;  hence,  the 
bank  discount  is  $9.30,  and  tlie  avails  $600  —  $9.30  =  $590.70. 

RULE. 

To  find  the  bank  discount  of  a  note  tluit  does  not  bear 
interest : 

1.  Find  the  interest  on  the  note  for  the  time  that  it  is 
to  run. 

2.  To  find  the  avails,  subtract  the  discount  from  the 
face  of  the  note. 

Note.  Remember  that  the  sum  discounted  is  the  amount  to  be  paid 
at  the  maturity  of  the  note  ;  hence,  if  the  note  bears  interest,  first  find  the 
amount  of  principal  and  interest. 


Similarly 

solve  the  following : 

Face  of  Note. 

Time  Discounted. 

Rate  of  Discount. 

1. 

$825.00 

60  days 

6% 

2. 

$927.18 

30    " 

5% 

3. 

$264.83 

45     " 

5% 

4. 

$169.47 

90     " 

7% 

5. 

$2,968.51 

4  months 

H% 

6. 

$417.80 

5       " 

8% 

7. 

$361.28 

6 

7% 

8 

$248.65 

50  days 

7% 

9. 

$1,827.90 

40     " 

6i% 

10. 

$83.70 

1  year,  3  months 

6% 

APPLICATIONS   OF  PERCENTAGE.  287 

11.    $850.  Buffalo,  N.  Y.,  July  21,  1896. 

Three  months  after  date,  for  value  received,  1  promise  to 

pay  to  Katharine  Sedgwick,  or  order,  Eight  Hundred  Fifty 

Dollars,  at  the  City  Bank. 

WiLLiSTON  Cook. 

Discounted  Aug.  15,  1896,  at  7%. 

12.  $1,200.  Philadelphia,  Pa.,  Aug.  1,  1896. 
Four  months  after  date  I  promise  to  pay  to  Richard  M. 

Johnson,  or  order.  Twelve  Hundred  Dollars,  value  received. 

Thojias  R.  Williams. 
Discounted  Oct.  15,  1896,  at  6%. 

13.  $128.50.  Detroit,  Mich.,  July  10,  1896. 
Sixty  days  after  date,  for  value  received,  we  promise  to 

pay  to  Henry  R.  Sunderland,  or  order.  One  Hundred  Twenty- 
eight  and  ^o"a  Dollars. 

Arthur  G.  Hunting. 

Peter  T.   Small. 
Discounted  Aug.  1  at  8%.     Count  days  of  grace. 

14.  S480.  Boston,  Mass.,  Sept.  1, 1896. 
Three  months  after  date,  for  value  received,  I  promise  to 

pay  to  Samuel  S.  Huston,  or  order,  Four  Hundred  Eighty 
Dollars,  with  interest  at  6  per  cent  per  annum. 

John  L.  Whitney. 
Discounted  Sept.  18  at  8%.     Count  days  of  grace. 

15.  $1,540.  Aurora,  III.,  Aug.  21,  1896. 
Two  months  after  date,  for  value  received,  I  promise  to 

pay  to  Joseph  H.  Freeman,  or  order,  Fifteen  Hundred  Forty 

Dollars. 

James  S    Campbell. 

Discounted  Sept.  1,  at  6^%. 


288  NEW  ADVANCED  ARITHMETIC. 

16.  8875.00.  Chicago,  July  10,  1896. 
Three  months  after  date,  for  value  received,  I  promise  to 

pay  to  the  Chemical  National  Bank  Eight  Hundred  Seventy- 
five  Dollars,  with  interest  at  7%  per  annum  after  maturity. 

Robert  S.  Daniel. 
Discounted  at  7%,  July  10. 

17.  $760.  Bloomington,  III.,  Aug.  3,  1896. 
Four  months  after  date,  for  value  received,  I  promise  to 

pay  to  the  National  State  Bank,  or  order.  Seven  Hundred 
Sixty  Dollars,  with  interest  at  seven  per  cent  per  annum 
after  due. 

James  L.  Atwood. 
Discounted  at  7 %  Aug.  3. 

317.  Mr.  A,  desiring  to  pay  a  debt  of  8460,  went  to  a 
bank  to  obtain  the  monej'.  Since  he  must  receive  8460  from 
the  bank,  it  is  evident  that  he  must  make  his  note  for  more 
than  that  amount.  He  wishes  to  borrow  the  money  for  4 
months.  The  current  rate  of  interest  is  7%.  If  he  should 
make  his  note  for  81-00,  the  banker  would  give  him  the  dif- 
ference between  81.00  and  the  interest  on  it  for  4  months 
and  3  days,  if  days  of  grace  are  counted.  The  interest  on 
$1.00  at  7%  for  4  months  and  3  days  is  23^-  mills.  The 
avails  of  such  a  note  would  be  81.00  —  80.023}i  =  80.97(;yi5. 
In  order  that  he  shall  receive  8460  his  note  must  be  made 
for  as  many  dollars  as  there  are  times  80.976y\  in  460. 
460  -i-  .976^5  =  471.27;  hence,  the  note  must  be  made  for 
«471.27. 

Proof.  The  interest  .on  $471.27  for  4  months  and  3  days 
at  7%  is  811.27+;  hence,  the  avails  of  such  a  note  would 
be  8460. 

Note.  In  calculating  interest  on  $1.00  for  short  periods,  the  second 
6%  method  is  most  convenient. 

From  the  foregoing  analysis  we  may  form  a 


APPLICATIONS   OF  PERCENTAGE.  289 

RULE. 
To  find  the  face  of  a  note  that  will  yield  a  given  amount 
Jyy  bank  discount: 

1.  Find  the  avails  of  a  note  for  $1.00  for  the  given  time 
and  at  the  given  rate,  and 

2.  IHvide  the  given  amount  by  it. 

Observe  that  there  are  two  kinds  of  problems  in  bank 
discount. 

(a)  To  find  the  avails  of  a  note  when  face,  time,  and  rate 
are  given. 

(h)    To  find  the  face  when  avails,  time,  and  rate  are  given. 

PROBLEMS. 

For  what  must  a  note  be  drav  ^  1       'eld : 


Amount. 

Time. 

Rate  of  Interest 

1. 

6724.50 

48  days 

6% 

2. 

$826.40 

60     " 

5% 

3. 

$692.24 

21     " 

8% 

4. 

$5,860 

90     " 

7% 

5. 

$84 

6  mouths 

6% 

6. 

$951.28 

4       " 

9% 

7. 

$10,864.80 

5       " 

8% 

8. 

$5,712.40 

80  days 

7% 

9. 

$438.76 

50     " 

6% 

10. 

$11,-372.12 

48     " 

5% 

318. 

EXCHANGE. 

1.  A,  in  Normal,  111.,  desired  to  pay  Wm.  Dulles,  Jr.,  in 
New  York,  $100.  It  would  not  be  safe  to  send  the  money  in 
a  letter  To  send  it  by  express  would  be  expensive.  This 
is  what  he  did.  He  went  to  the  bank  and  purchased  an 
order  upon  another  bank  in  New  York  to  pay  the  $100  to 
the  order  of  Wm.  Dulles.  [See  next  page.]  The  Normal 
bank  keeps  money  on  deposit  in  the  New  York  bank,  and 


290 


NEW  ADVANCED   ARITHMETIC. 


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APPLICATIONS   OF  PERCENTAGE.  291 

sells  its  orders  to  people  ■who  wish  to  pay  bills  in  Eastern 
cities. 

2.  A  Draft  is  an  order  made  by  one  party  upon  a  se^onit 
to  pay  a  designated  sum  to  the  order  of  a  third  party. 

3.  The  party  issuing  the  draft  is  the  Maker,  the  party 
upon  whom  the  order  is  drawn  is  the  Drawee,  and  the  party 
in  whose  favor  it  is  drawn  is  the  Payee. 

4.  In  tlie  draft  shown  on  the  preceding  page,  W.  H. 
Schureman  &  Co.  is  the  maker;  United  States  National 
Bank,  the  drawee;   and  ^Ym.  Dulles,  .Jr.,  the  payee. 

5.  Exchange  is  a  process  of  paying  obligations  due  at 
distant  places  by  transmitting  drafts. 

6.  The  terms  par^  discount,  and  premium  are  applied  to 
drafts  in  the  same  way  as  to  bonds. 

319.      PROBLEMS. 

I.  What  is  the  cost  of  a  draft  on  Xew  York  for  $fi50, 
when  exchange  is  at  i%  premium? 

What  is  the  cost  of  a  draft  for : 

2.  $185.40  on  Chicago  at  -^-q7c  premium? 

3.  $1:67.32  on  New  Orleans  at  i%  premium? 

4.  $1,875.12  on  Boston  at  ^%  premium? 

5.  $2,61)4.38  on  Pittsburg  at  ^-^%  discouut? 

6.  $10,850  on  Philadelphia  at  ^%  premium? 

7.  $791.13  on  New  York  at  i%  discount? 

8.  $2,874.93  on  St.  Louis  at  ,]-%  discount? 

9.  $167.91  on  Cincinnati  at  ^%  premium? 

10.    $16,824.57  on  San  Francisco  at  i%  premium? 

II.  If  exchange  is  at  j'^  %  premium,  how  large  a  draft  will 
$1,001  buy? 

AxALTSTS.  If  exchange  is  at  ^5^%  premium,  a  draft  for  Sl.OO  will  cost 
$1,001.  $1,001.00  will  \my  a  draft  whose  face  i;s  as  mauy  times  $1.00  as 
$1,001.00  is  times  $1,001. 


292  NEW  ADVANCED  ARITHMETIC. 

12.  If  exchange  is  ^%  discount,  how  large  a  draft  will 
$467.50  buy?  $328.37?  $572.20? 

13.  How  large  a  draft  will  $5,824.75  buy,  if  exchange  is 
1%  premium?  |%  discount?  i%  premium?  §%  discount? 

14.  How  large  a  draft  will  $325o50  buy,  if  exchange  is 
1%  disccunt?  |%  discount'^  1^%  premium?  §%  premium? 

15.  Find  the  cost  of  a  draft  for  $650.40,  at  1|%  premium ; 
at  1§%  discount,  at  ^%  premium,  ati%  discount. 

320.  1  The  drafts  with  which  we  have  been  dealmg  are 
called  Sight  Drafts,  because  payable  when  presented  A 
second  form  is  sometimes  used,  called  Time  Drafts,  payable 
a  specified  time  "  after  sight;  "  that  is,  after  they  have  been 
shown  to  the  drawee  and  he  has  stamped  "■  accepted,"  with 
the  date  of  acceptance,  and  has  written  the  name  of  the 
drawee  on  the  face, 

2.  Time  drafts  differ  from  sight  drafts  in  no  essential 
respect  except  in  the  designation  of  the  time  which  they  are 
to  run.  If  the  words  "Thirty  days  after  sight"  preceded 
*'  Pay  to  the  order  of,"  in  the  draft  given,  it  would  be  a  time 
draft. 

3.  It  is  evident  that  such  a  draft  would  not  ordinarily  be 
at  par,  since  the  purchaser  cannot  have  it  cashed  by  the 
drawee  until  its  maturity.  If  sight  exchange  is  at  par,  such 
a  draft  is  worth  its  face,  less  bank  discount  for  the  time  it 
is  to  ruuo 

PROBLEMS. 

1.  What  is  the  value  of  a  90-day  draft  for  $800,  ex- 
change being  ^%  premium,  and  the  current  rate  of  interest 
being  6  %  ?     ( Witli  grace.) 

Analysis.  The  interest  on  $800  for  93  days,  at  6%,  is  $12.80.  It  ex- 
change were  at  par,  the  draft  would  be  worth  $800  -  $12.80  =  $787.20. 
Since  exchange  is  at  ^%  premiuni,  it  is  worth  \%  of  $800  +  $787.20. 

2.  Find  the  cost  of  a  45-day  draft  for  $460.50,  exchange 
being  ^%  discount,  and  interest  at  7%. 


APPLICATIONS   OF  PERCENTAGE.  293 

3.  AVhat  is  the  face  of  a  90-day  draft  that  can  be  pur- 
chased for  §1,175.62,  if  exchange  is  at  1%  discount,  and  the 
rate  of  interest  is  7%  ?     (With  grace.) 

Analysis.  If  exchange  were  at  par,  a  90-day  draft  for  §1  00  would 
cost  $1.00  —  $0  OlSjJj.  Siuce  exchange  is  at  i  %  discount,  there  should  be 
a  further  withdrawal  of  $.003^;  heuce,  a  draft  for  SI. 00  would  cost  $1.00 
—  $0.02Iy\  =  $0.978iV-  $1,175.62  will  buy  a  draft  whose  face  is  as  many 
times  $1.00  as  1,175.62  is  times  >!0.978iV- 

Note.     Change  divisor  and  dividend  to  twelfths  for  perfect  accuracy. 

4.  How  large  a  draft,  payable  in  4  months,  exchange  ^% 
premium,  interest  at  8%,  can  be  bought  for  §1,250? 

5.  Find  the  cost  of  a  30-day  draft  for  §427.50,  exchange 
at  \%  discount,  iiiterest  at  8%  . 

6.  Find  tlie  cost  of  a  sight  draft  for  $784.90,  at  \% 
premium. 

7.  Exchanged  the  preceding  draft  for  a  45-day  draft,  ^% 
discount,  interest  6% .     What  was  the  face  of  the  time  draft? 

8.  A  commission  merchant  sold  10,000  bushels  of  wheat 
at  87  cents,  withdrew  his  commission  of  1^%,  and  with  the 
remainder  bought  a  60-day  draft,  exchange  \%  premium, 
interest  at  6%.     What  was  its  face? 

9.  A  farmer  sold  120  acres  of  land  at  §65  an  acre,  re- 
ceiving from  the  purchaser  his  note  for  4  months  without 
interest.  He  at  once  discounted  the  note  at  a  bank  at  7%, 
and  with  the  proceeds  bought  a  time  draft  on  New  York, 
due  in  4  months,  exchange  being  1%  discount,  interest  6%. 
If  he  received  the  face  of  the  draft,  did  he  gain  or  lose  by 
the  deal?     How  much? 

When  drafts  are  paid  in  the  country  in  which  they  are 
drawn,  the  exchange  is  called  Domestic,  or  Inland.  The 
system  is  extended  also  to  foreign  countries,  and  constitutes 

321.    FOREIGN    EXCHANGE. 

1.  Drafts  drawn  upon  banks  in  foreign  countries  have 
their  face  value  expressed  in  the  currency  of  that  country. 


294  NEW  ADVANCED  ARITHMETIC 

2.  When  a  system  of  exchange  is  established  jetween 
two  countries  it  is  necessary  to  be  able  to  express  the  value 
of  the  currency  of  each  in  the  currency  of  the  other.  Such 
an  expression  is  called  the  par  of  exchange.  The  legal 
value  of  a  pound  sterling  is  $4.8665.  If  exchange  varies 
from  this  price,  it  is  above  or  below  par.  Bills  of  exchange 
on  England  are  called  Sterling  Bills.  Their  price  is  quoted 
at  the  cost  of  a  pound  sterling  in  United  States  money. 

3.  Foreign  bills  of  exchange  are  sometimes  issued  in  sets 
of  three,  called  respectively  the  first,  second,  and  third  of 
exchange.  Such  bills  are  transmitted  at  different  times  to 
avoid  losses  and  delays.  The  one  reaching  its  destination 
first  is  paid,  and  the  others  become  void. 

4.  The  following  is  the  more  common  form  used: 
Exchange  for 

£250.  Chicago,  III.,  June  1,  1892. 

On  demand  of  this  Bill  of  Exchange  pay  to  the  order  of 
Dillon  Bros.  Two  Hundred  Fifty  Pounds  sterling,  value  re- 
ceived, and  charge  the  same,  as  per  advice,  to 

BuowN  Bros. 
To  Dunn  &  Co., 

London. 
No.  185. 

5.  Foreign  bills  may  be  payable  upon  presentation,  or 
after  a  specified  time  The  value  of  any  foreign  exchange 
is  now  quoted  as  that  of  other  commodities,  at  a  specified 
price  per  pound  sterling,  per  franc,  etc.  The  following  is 
the  Chicago  quotation  for  Feb.  15,  18'J2  : 

For  One  Pound  Sterling  $1.!)0 

One  F"'ranc  on  France  A'i)k 

One  Franc  on  Belgium  .19^ 

One  Franc  on  Switzerland     .19^ 
One  German  Mark  .24^ 

One  HoUandish  Florin  .401 


APPLICATIONS  OF  PERCENTAGE.  295 

One  Austrian  Florin  .42 

One  Lira  Italian  .19^ 

One  Rouble  .50 

One  Finnish  Mark  .19| 

One  Krona  on  .Sweden  .27 

One  Krone  on  Denmark  .27 

One  Krone  on  Norway  .27 

PROBLEMS. 

At  the  quotations  given  find  the  cost  of  each  of  the 
following  bills  of  exchange : 

1.  For  456  German  marks.  3.   For  1,283  roubles. 

2.  For  874  Austrian  florins.  4.    For  697  lire. 

5.  For  384  pounds  sterling. 

6.  How  large  a  draft  on  Paris  can  be  bought  for  S245.70? 
7. .  On  Brussels  for  $174.33? 

8.  On  London  for  $573.30? 

9.  What  is  the  rate  when  a  draft  on  Loudon  for  £150 
costs  $732? 

10.  What  is  the  rate  when  a  draft  on  Paris  for  1200 
francs  costs  $232  50? 

11.  When  a  draft  on  Vienna  for  800  florins  costs  $334? 

12.  When  a  draft  on  Helsingfors  for  640  marks  costs 
$121.60? 

322.    EQUATION  OF  PAYMENTS. 

The  Equation  of  Payments  is  the  process  of  finding  a 
time  at  which  several  sums  due  at  different  times,  and  not 
bearing  interest,  can  be  paid,  without  loss  to  debtor  or  cred- 
itor, and  without  the  transfer  of  money  for  interest. 

If  the  several  sums  bear  interest,  all  could  be  paid  at  any 
time  by  chscharging  the  principal  and  accrued  interest,  and 
no  one  would  lose. 


296  NEW  ADVANCED  ARITHMETIC. 

The  principle  upon  which  the  operations  are  based  is  very 
simple.  If  money  is  paid  before  it  is  due,  interest  should 
be  allowed  upon  it ;  if  not  paid  until  after  it  is  due,  it  should 
bear  interest  from  maturity  until  date  of  payment. 

Illuslrative  Problem.  I  owe  SoOO  due  in  4  months;  S600 
due  in  seven  months;  and  Si, 000  due  in  nine  months:  what 
is  the  average  term  of  credit? 

These  sums  do  not  bear  interest  until  after  maturity.  If 
any  one  of  them  should  be  paid  before  its  maturity  the 
debtor  would  lose  the  use  of  the  mone}'  for  the  remaining 
time.  If  it  should  not  be  paid  until  after  maturit}^,  tlie 
creditor  would  lose  interest. 

Assume  money  to  be  worth  6%  interest,  and  the  several 
debts  to  be  paid  to-day.  On  the  first  the  debtor  would  lose 
the  interest  for  four  months,  which  is  810.  On  the  second 
he  would  lose  interest  for  seven  months,  or  S21 ;  and  on 
the  third,  §45.  Consequently,  if  the  debts  were  paid  in 
full  to-day,  the  debtor  would  lose  S76.  He  may  pay  them 
together  T>-ithout  loss  by  keeping  them  long  enough  after 
to-day  to  earn  S76.  The  interest  on  §2,100  for  one  day 
at  6%  is  35  cents.  §76  H-  .35  ==  217+.  They  should  be 
paid  218  days  from  to-day. 

This  method  of  finding  the  time  of  paj^ment  is  called  The 
Interest  Method.  The  time  sought  is  called  the  Equated 
Time.     It  may  be  found  also  by 

THE    PRODUCT    METHOD. 

The  interest  on  S500  for  4  months  equals  the  interest  on 
$2,000  for  1  month.  The  interest  on  8000  for  7  months 
equals  the  interest  on  84,200  for  1  month.  The  interest  on 
81,000  for  9  months  equals  the  interest  on  89,000  for 
1  month.  The  interest  on  the  several  amounts  equals  the 
interest  on  §15,200  for  1  month.  But  the  interest  on 
815,200  for  1  month  equals  the  interest  on  82,100  for  as 
many  months  as  §15,200  is  tinios  §2,100.      §15,200  is  7j\ 


APPLICATIONS   OF  PERCENTAGE.  297 

times   $2,100;    hence,    the   equated   or   average   tune   is  7 
months  aud  8  days. 

PROBLEMS. 

1.  Find  the  average  time  for  the  payment  of  $325  due  in 
30  days;  $650  due  in  40  days;  $500  due  in  60  days;  and 
$825  due  in  90  days ;  all  without  interest. 

2.  Find  the  equated  time  for  the  payment  of  $275  due  in 
3  months;  $580  due  in  5  months;  $1,020  due  in  7  months; 
$1,260  due  in  10  months;  no  interest  being  charged  on  any 
account. 

3.  Find  the  equated  time  for  the  payment  of  the  balance 
of  a  debt  of  $1,250  which  was  to  run  one  year,  without 
interest,  one  half  of  it  having  been  paid  at  the  end  of 
3  months ;  one  fourth  of  the  remainder  at  the  end  of  6 
months ;  aud  one  fourth  of  that  remainder  at  the  end  of 
9  months. 

4.  When  could  the  following  non-interest-bearing  debts 
be  paid  at  one  time  without  loss  ? 

$840  due  May  1;  $650  due  June  15;  $900  due  July  8; 
$1,275  due  August  1. 

Note.     Assume  May  1  as  the  date  of  payment. 

5.  On  March  1,  1888,  A  owed  B  $2,500  due  December  15, 
without  interest.  On  June  12  he  paid  $500,  and  September 
10,  $1,000.  When  should  he  pay  $400  to  entitle  him  to 
keep  the  remainder  until  Dec.  15,  1889? 

6.  What  was  the  equated  time  for  paying  the  following 
non-iuterest-bearing  bills  ?  _ 

March  1,  1889,  a  bill  of  $300  for  60  days. 
April  15,  1889,     "     "    $400  for  30  days. 
June  10,  1889,      "     "    $583.50  for  4  months. 
July  1,  1889,         "     "    $962.80  for  3  months. 

7.  Jan.  1,  1890,  A  bought  a  bill  of  goods  amounting  to 
$2,560,  on  90  days'  time,  without  interest.     On  January  16, 


298  NEW  ADVANCED  ARITHMETIC, 

he  paid  $850 ;  on  February  1,  $725.  If  he  settled  the  bill 
at  maturity,  liow  much  should  the  balance  be  discounted, 
money  being  worth  6  %  ? 

8.  A  man  owes  a  bill  of  $1,200  due  in  8  months,  without 
interest.  How  much  must  he  pay  at  the  end  of  4  months, 
to  extend  the  balance  2  months? 

9.  Find  the  average  time  of  payment  for  the  following 
bills,  which  do  not  bear  interest. 

January  1,  $400  on  3  months. 
February  15,  $600  on  30  days. 
May  10,  $560  on  4  months 
June  12,  $800  on  60  days. 
June  20,  $250  on  20  days 

10.  A  merchant,  on  the  first  cf  March,  bought  goods  1x) 
the  amount  of  $1,000.  He  agreed  to  pay  $250  cash,  $250 
on  the  3d  of  May;  $250,  July  4;  and  $250,  Sei)tember  15, 
all  without  interest.  He  prefers  to  pay  the  whole  at  one 
time  ;  when  should  it  be  ? 

11.  William  Jones  of  Council  Bluffs  buys  goods  of  Mar- 
shall Field  as  follows : 

1.  May    1,  bill  of  $.300  on  3  mo.  credit. 

2.  May  15,      "      $800  "  4  mo.     '•' 

3.  June    1,       "      $500  "  6  mo.      " 

4.  June    9,      "      $900  for  cash. 

Marshall  Field  agrees  to  take  Mr.  Jones's  note  for  the 
whole  amount,  for  30  days,  with  interest.  When  should 
the  note  be  dated? 

XoTK  1.  First  find  the  equated  time  of  payment.  Assume  for  this 
purpose  the  earliest  date  on  which  a  payment  falls  due. 

(1)3  mo.  credit  from  May    1.  Due  .\u<just  1. 

(2)  4  mo.       "       "        May  15.  "     September  1.5. 

(3)  6  mo.       "       "       June    1.  "     Decciinber  1. 

(4)  Cash  payment.  "    Juue  9. 
June  9  to  August  1  =  53  days,  etc. 

Note  2.     For  further  w^rk  see  Appendix. 


APPLICATIONS  OF  PERCENTAGE.  299 

323.      MISCELLANEOUS    PROBLEMS. 

1.  A  note  for  $428.50,  dated  Aug.  15,  1889,  is  to  run  90 
days,  without  interest.  It  was  discounted  October  1,  at  8% . 
Wiiat  were  the  proceeds? 

Note.  The  following  illustration  explains  the  method  of  finding  the 
date  of  maturity  of  a  note. 

A  note  is  dated  May  18,  and  is  to  run  72  days : 

May 13  days. 

June     ....     30  days. 
July      ....     31  days. 
Matures,  Aug 1  day. 

75  days. 

2.  Find  the  difference  between  the  true  discount  and  the 
bank  discount  of  the  above  note  if  discounted  August  15, 
at  6  % . 

Note.  True  discount  is  interest  on  what  ?  Bank  discount  ?  Wliat 
is  the  difference,  then  ? 

3.  A  merchant  bought  a  bill  of  goods  for  $1,875.60,  on  90 
days'  time.  Finding  that  he  could  get  a  discount  of  5%  of 
the  whole  amount  by  paying  cash,  he  borrowed  the  requisite 
amount  from  a  bank.  For  what  must  he  draw  a  note  for  90 
days,  current  rate  being  8%,  to  pay  the  bill?  How  much 
did  he  gain  by  the  process?     (Bank  Discount.) 

4.  A  merchant  bought  a  bill  of  goods  for  $1,674.20.  He 
was  allowed  5%  off  for  cash.     What  did  he  pay? 

5.  A  stock  train  has  29  cars,  each  containing  19  cattle, 
whose  average  weight  is  1,450  pounds.  They  sell  for  $5.25 
a  hundred.     What  do  they  bring? 

6.  A  man  bought  a  half -section  of  land  at  $81.50  an  acre. 
He  gave  one  note  for  $12,650,  a  second  note  for  $5,830,  paid 
$3,600  in  cash,  and  gave  a  third  note  for  the  remainder. 
What  was  the  face  of  the  third  note  ? 

.7.  The  above  notes  were  all  dated  July  1,  1892.  The  first 
was  due  in  2  vears  and  bore  6%   interest.     Falling  heir  to 


J^ 


300  NEW  ADVANCED  ARITHMETIC. 


discount.     What  did  he  pay  for  it? 

8.  The  second  of  the  above  notes  was  due  in  3  years,  and 
bore  interest  at  6%.  On  Sept.  25,  having  sold  his  wheat 
crop,  he  discounted  the  second  note  by  true  discount  at  8%. 
What  did  he  pay  for  it? 

9.  The  third  of  the  above  notes  was  due  in  4  years,  and 
bore  annual  interest  at  5  % .  He  paid  no  interest  until  the 
maturity  of  the  note.     What  amount  was  then  due? 

10.  He  borrowed  the  $3,600,  in  Problem  6,  at  a  bank  by 
giving  his  note  for  4  months  at  7  %  .  What  was  the  face  of 
his  note  which  yielded  him  that  amount? 

11.  Find  the  amount  of  the  following  bUl : 

24  Arithmetics  @  95^ 
16  Readers  @  81.20. 
20  Geographies  @  Si. 40. 
18  Grammars  @  75«<. 
42  Spellers  @  22<''. 
The  dealer  made  a  discount  of  33  \%  from  the  regular  price, 
and  a  further  discount  of  5%  of  the  balance  for  cash. 

12.  Bought  a  bill  of  8748.80,  the  seller  making  a  discount 
of  33^%  and  5%  .     What  was  the  amount  to  be  paid? 

13.  What  is  the  weight  of  a  stone  roller  8  feet  long  and  8 
feet  in  circumference  (tt  =  -2^^),  the  stone  being  2.3  times  as 
heavy  as  water? 

14.  What  is  the  weight  of  a  grindstone  4  inches  thick  and 
30  inches  in  diameter,  the  hole  being  2  inches  in  diameter, 
the  stone  being  2.143  times  as  heavy  as  water? 

15.  What  fraction  is  that  which  being  multiplied  by  §  of 
I  gives  5  of  j\  as  a  product  ? 

16  Bought  250  yards  of  broadcloth  for  £87  10  s.  The 
import  duty  was  36%.  Transportation  and  other  charges 
amounted  to  812.50.     It  was  sold  at  83.25  a  yard.     What 


APPLICATIONS   OF  PERCENTAGE.  301 

was  the  per  cent  of  gain,  the  pound  sterling  being  worth 
$4.90?     (Approximate.) 

17.  Bought  a  piano  for  $560  on  March  5,  1895.  The 
agreement  is  that  I  shall  pay  6%  interest  on  the  purchase 
price  with  the  privilege  of  making  payments  upon  which  in- 
terest shall  be  allowed  at  the  same  rate.  I  made  the  follow- 
ing payments  :  May  25,  $175  ;  July  18,  $160  ;  Nov.  12,  $125. 
What  was  due  March  5,  1896? 

18.  Find  the  cost  of  a  draft  on  London  to  pay  for  the 
broadcloth  in  Problem  16,  ey change  being  $4.89  for  a  pound 
sterling. 

19.  A  man  bought  a  business  lot  60'  X  160'  at  $4i  a  square 
foot.  He  erected  a  six-story  building,  costing  $50,000,  on 
the  lot.  In  the  first  story  there  are  two  store-rooms,  which 
rent  for  $1,500  each.  There  are  20  rooms  on  each  of  the 
other  floors.  Those  on  the  second  floor  average  $20  a  month. 
There  is  a  reduction  of  $3  a  month  as  the  stories  ascend.  If 
insurance,  taxes,  and  care  of  the  building  amount  to  $3,500, 
what  interest  will  he  receive  on  his  investment? 

20.  If  H%  be  paid  for  collecting  the  rents  in  the  above 
building,  what  is  the  collector's  commission  per  month? 

21.  What  is  the  compound  interest  on  $345  for  4  yr.  3  mo. 
20  d.,  at6%? 

22.  What  time  will  be  required  for  $460  to  earn  $64.80,  at 
7%? 

23.  What  principal  earns  $56.40  interest  in  2  yr.  5  mo. 
15  d.,  at  6%? 

24.  Divide  .46083  by  .37. 

25.  What  is  the  difference  between  London  local  time  and 
New  York  local  time  ?  If  a  cablegram  be  sent  from  London 
at  8  A.  M.,  and  one  hour  be  employed  in  getting  it  to  its  des- 
tination in  New  York,  at  what  time  will  it  reach  there  ? 

26.  Bought  5%  school  bonds  at  102.  What  rate  of  inter- 
est on  the  investment  do  they  pay? 


302  NEW  ADVANCED  ARITHMETIC. 

27.  Bought  40  shares  of  National  Bank  stock  at  160. 
They  pay  a  semi-annual  dividend  of  $160.  What  is  the 
rate  of  dividend?     What  rate  do  they  pay  on  their  cost? 

28.  What  is  the  diameter  of  your  bicycle  wheels?  How- 
many  revolutions  do  they  make  in  going  a  mile?  How  many 
revolutions  do  they  make  for  one  revolution  of  a  pedal? 
How  many  revolutions  of  the  pedals  will  take  the  wheel  one 
quarter  of  a  mile? 

29.  What  is  the  "accurate"  interest  on  $560.40  for  9a 
days  at  6%  ? 

30.  What  principal  will  amount  to  $1,200  in  3  yr.  7  mo 
21d.,  at8%? 

31.  I  send  to  a  commission  merchant  $1,000  to  be  in- 
vested in  wheat  at  57|  cents  a  bushel,  after  deducting  his. 
commission  at  2%.  How  many  bushels  will  my  remittance 
buy? 

32.  I  have  a  bin  36  feet  long,  10  feet  wide,  and  9  feet 
high,  which  is  filled  with  shelled  corn.  A  commission  mer- 
chant sells  it  for  me  at  24|  cents  a  bushel,  charging  1^% 
commission,  and  remits  to  me  the  balance?  What  is  the 
amount  of  his  remittance? 

33.  I  own  80  shares  of  Building  and  Loan  stock  upon 
which  I  pay  50  cents  a  share  monthly.  After  paying  for  4 
years  I  draw  out  my  investment  and  receive  $936.  What 
rate  of  interest  do  I  receive? 

Note.    Find  average  time. 

34.  A  man  whose  watch  shows  Chicago  local  time  finds 
that  it  is  36  min.  30  sec.  faster  than  the  local  time  of  the 
place  in  which  he  is.     What  is  his  longitude?     (See  Table.) 

35.  In  a  school  of  750  pupils  the  number  of  boys  is  87^%^ 
of  the  number  of  girls.  How  many  boys  are  there  in  the 
school  ? 

36.  How  many  prescriptions  each  weighing  5  dr.  1  so. 
12  gr.  can  be  made  from  1  lb.  7  oz.  2  dr.  2  sc.  16  gr.  ? 


APPLICATIONS   OF  PERCENTAGE.  303 

37.  The  tax  for  State  purposes  in  Illinois  is  53  mills  on  3 
hundred  dollars.  What  does  this  amount  to  on  an  assessed 
valuation  of  Si 5, 624?  ^ 

•  38.    Find  the  cost  of  the  following  at  S17.50  a  thousand: 
24  studs  2  X  4,  18  feet  long. 
32  joists  2  X  10,  16  feet  long. 
8  sills  8  X  10,  14  feet  long. 
1520  feet  common  fencing. 

39.  An  auctioneer  sold  the  following  articles  at  a  farmer's 
sale.     What  is  his  commission  at  2%  ? 

5  horses,  averaging  S65. 
8  cows,  averaging  $36.50. 

2  wagons,  one  for  $18.75,  the  other  for  $31.60. 
20  tons  hay  at  $9.75. 

3  plows  at  $5.31. 

1  harrow  at  $4.90. 
26  hogs  at  $8.30. 

40.  At  the  above  sale  a  discount  of  5%  was  made  for 
cash.  What  amount  would  discharge  the  obligation  of  the 
man  who  bought  the  horses  and  the  cows  ? 

41.  It  was  a  condition  of  the  above  sale  that  purchasers 
might  pay  in  one  year  without  interest.  The  man  buying 
the  wagons  and  the  hay  discounted  his  note  at  8%  — true 
discount  —  at  the  end  of  4  mo.  18  d.     What  did  he  pay? 

42.  The  man  buying  the  hogs  borrowed  the  money  from 
a  bank,  at  7%,  for  90  days,  with  grace,  and  paid  cash  for 
them.     What  was  the  face  of  his  note  ? 

43.  Only  what  common  fractions  can  be  changed  to  pure 
decimals?     Why?     Change  j^  to  a  decimal  (four  places), 

44.  Find  the  compound  interest  on  $465.72  for  3  yr. 
2  mo.  8  d.,  at  ^%. 

45.  Reduce  ||^  X  ^-^  -.-  |  of  |  of  /j. 


304  NEW  ADVANCED  ARITHMETIC. 

46.  In  what  time  will  a  principal  of  $864.12  amount  to 
$1,040  at  7%  simple  interest? 

47.  What  principal  will  amount  to  $501.83  in  10  yr.  3  mo. 
15  d.,  at  7%  compound  interest? 

48.  The  time  from  9  o'clock  to  15  min.  20  sec.  past  10 
is  what  part  of  a  day  ? 

49.  What  is  the  area  of  a  city  lot  50'  X  150'?  What  is 
it  worth  at  $6,000  an  acre? 

50.  Add  21  lb.  8  oz.  12  pwt.  21  gr. ;  12  lb.  10  oz.  16  gr. ; 
5  lb.  18  pwt.  19  gr. ;  26  lb.  8  pwt.  22  gr. ;  10  oz.  3  pwt. 
9gr. 

51.  What  is  the  date  of  your  birth?  How  old  are  you 
to-day  ? 

52.  Change  -^^g  of  a  square  mUe  to  integers  of  lower 
denominations. 

53.  Change  .325  of  a  mile  to  integers  of  lower  denomi- 
mations. 

54.  An  auctioneer's  commissions  for  a  year,  at  2i%, 
amounted  to  $3,124.80.  He  was  employed  265  days.  What 
was  the  daily  average  of  his  sales? 

55.  A  manufacturer  received  from  his  foreign  agent  50 
bales  of  wool,  250  pounds  each,  invoiced  at  36  cents  per 
pound,  and  36  bales,  300  pounds  each,  invoiced  at  31  cents 
a  pound.     What  was  the  duty,  at  25  %  ad  valorem? 

56.  The  width  of  this  book  is  what  per  cent  of  its  length? 

57.  The  area  of  this  page  is  what  decimal  of  a  square 
yard? 

58.  What  will  it  cost  to  carpet  this  room  with  Brussels, 
I  of  a  yard  wide,  at  $1.12  a  yard,  if  the  strips  run  length- 
wise, with  no  loss  for  matching?  Would  the  cost  be  changed 
if  the  strips  ran  crosswise,  with  no  loss  for  matching? 

59.  If  15  horses  eat  101  :^  bushels  of  oats  in  12  days,  how 
many  bushels  will  32  horses  eat  in  21  days  at  the  same  rate? 


APPLICATIONS  OF  PERCENTAGE.  305 

60.  A  and  B  can  do  a  piece  of  work  in  9  days,  and  A  can 
do  it  in  16  days.  If  they  receive  $32  for  the  job,  and  each 
is  to  be  paid  in  proportion  to  what  he  does,  what  should  B 
receive? 

61.  Three  boys  start  from  the  same  point,  at  the  same 
time,  and  ride  their  bicycles  around  a  block  in  the  same  direc- 
tion. The  block  is  ^  of  a  mile  on  a  side.  If  the  first  rides 
at  an  average  rate  of  10  miles  an  hour,  the  second  12  miles, 
and  the  third  15  miles,  how  long  before  they  will  be  together 
at  the  starting-point? 

62.  A  man  wished  to  purchase  a  horse,  saddle,  and  bridle. 
A  dealer  offered  to  sell  him  a  horse  for  $160,  a  pony  for  ^  of 
what  he  asked  for  the  horse,  saddle,  and  bridle,  or  the  pony, 
saddle,  and  bridle  for  \  of  what  he  asked  for  the  horse. 
What  was  the  price  of  the  saddle  and  bridle  ?  of  the  pony  ? 

63.  On  the  Centigrade  thermometer  the  freezing-point  is 
0°  and  the  boiling-point  100°.  How  is  it  on  the  Fahrenheit 
thermometer?  What  is  the  relation  between  the  two  scales? 
Change  72°  Fahrenheit  to  the  Centigrade  scale. 


306  NEW  ADVANCED  ARITHMETIC. 


sectio:n"  IX. 

324.     RATIO. 

1  3  is  what  part  of  6?  4  is  what  part  of  7?  5  is  what 
part  of  6  ?  of  9  ?  of  15  ?     9  is  what  part  of  1 2  ?  of  7  ?  of  6  ? 

2.  h  is  what  part  of  f  ?  of  §?  of  §?  of  J?  of  ^? 

3.  U  is  what  part  of  88?  of  810?  of  816?  of  $2? 

4.  5  feet  is  what  part  of  10  feet?  of  12  feet?  of  7  feet? 
of  15  feet?  of  2  feet? 

5.  §  of  1  pound  is  what  part  of  f  of  1  pound  ?  of  |  of  1 
pound?  of  i  of  1  pound? 

6.  Instead  of  the  forms  used  above,  a  briefer  form  may  be 
used  to  express  the  same  relation.  When  a  colon  is  placed 
between  tu'o  numbers,  it  indicates  that  the  first  is  to  be  meas- 
ured by  the  second. 

7.  One  number  is  measured  by  another : 

(a)  By  finding  what  part  the  first  is  of  the  second;  or, 

(b)  By  finding  how  many  times  the  second  number  the 
first  is. 

This  is  done  b}"  dividing  the  first  number  by  the  second. 
8    Such  an  expression  is  called  a  Ratio. 
The  ratio  of  one  number  to  another  is  that  relation  which 
is  found  by  dividing  the  former  by  the  latter. 

9.  5  :  8  is  a  ratio.  It  is  read.  *■'  the  ratio  of  5  to  8."  Its 
value  is  ;|.  The  expression  may  also  be  read  "  5  is  f  of  8." 
Tiie  form  is  new,  but  the  idea  is  familiar. 

10.  Read  the  following  in  both  ways  :  5  :  10.  6:7.  i  :  §. 
$8:  $3.     5  lb.  4  oz.:  6  lb.  13  oz. 

11.  3  ft.  9  in.  :8  ft.  4  in.  7^:2^.  8.7:2.9.  5  A.  12 
sq.  rd.  :  15  A.  36  sq.  rd. 


PROPORTION.  307 

12.  The  first  term  of  a  ratio  is  the  Antecedent,  and  the 
second  term  the  Consequent. 

13.  A  ratio  is  like  a  fraction,  the  antecedent  correspond- 
ing to  the  numerator,  and  the  consequent  to  tlie  denominator. 
The  difference  lies  in  the  fact  that  the  antecedent  must  al- 
ways be  thought  of  as  a  part  of,  or  some  number  of  times, 
the  consequent. 

14.  Find  the  value  of  each  of  the  following  ratios :  7  :  2.1. 
?.|.  ^:^•  %-2^:%d\.  3  mi.  20  rd. :  2  mi.  5  rd.  .7:2.1. 
3  lb.  2  oz. :  5  lb.  14  oz. 

15.  The  terms  of  a  ratio  must  be  like  numbers.     Why? 

325.     PROPORTION 

1.  Compare  the  value  of  5  :  6  with  that  of  10:  12.  Of 
4 :  5  with  12  :  15.     Of  ^  :  f  with  g  :  |. 

2.  Find  a  ratio  equal  to  8:  12.  Equal  to  $7:  $21.  3^ 
pounds  :  2  pounds. 

3.  Find  a  number  whose  ratio  to  12  equals  6:9.  Whose 
ratio  to  |  equals  t^(j  :  j- 

4.6:7^-10:?     12:15  =  8:?     g  :  ?  =  4  :  12. 

Note.  6  :  7  =:  10  :  ?  is  read,  "the  ratio  of  6  to  7  =  the  ratio  of  10  to 
what  number  1  "  or,  "  6  is  the  same  part  of  7  that  10  is  of  what  number  ?  " 

5.  ?:  15  =  9:27.     6J-:?  =  9:27.      11:  33  =  ?:  19. 

6.  A  double  colon  is  usually  used  instead  of  the  sign  of 
equality.  Thus,  4  :  6  : :  8  :  12.  Such  an  expression  is  called 
a  Proportion. 

7.  A  proportion  is  an  equality  of  two  ratios. 

The  first  ratio  is  called  the  First  Couplet,  and  the  second 
ratio,  the  Second  Couplet. 

8.  The  first  and  last  terms  of  a  pro])ortion  are  called  the 
Extreme  Terms ;  the  second  and  third  terms,  the  Mean 
Terms. 


308  NEW  ADVANCED  ARITHMETIC. 


326.     PRINCIPLES. 

1.  The  product  of  the  extreme  terms  equals  the  product  of 
the  mean  terms. 

Peoof.     a  ratio  may  always  be  expressed  in  the  form  of  a  fractiou. 

Since  the  two  ratios  of  a  proportion  are  equal,  the  fractions  to  which 
they  are  equivalent  must  be  equal. 

If  two  equal  fractions  are  made  to  have  the  same  denominator,  their 
numerators  will  be  equal ;  hence,  the  product  of  the  first  numerator  and 
second  denominator  will  equal  the  product  of  the  first  denominator  and 
second  numerator. 

2.  If  the  product  of  the  extremes  be  divided  by  either 
meau,  the  quotient  will  be  the  other  mean. 

3.  If  the  product  of  the  means  be  divided  by  either  ex- 
'^reme,  the  quotient  ^vill  be  the  other  extreme. 

Prove  Principles  2  and  3. 

.Make  rules  for  finding  either  extreme  or  either  mean  of  a 
proportion. 

PROBLEMS. 

Find  the  missing  term  in  each  of  the  following  propor- 
tions : 

1.  6:  10  =  15:a;. 

2.  12:  39  =  a-:  91.  9.    8:  15::  a;:  24. 

3.  ^:x  =  ^:^.  10.    15:  36:  :  72  :  a-. 

4.  a;  125  =  72:  108.  11.    a; :  48  :  :  60:  75. 

5.  26A  .36A.-13  T.  :x.  12.    30  :  a;  • :  f  :  2^. 

6.  $625  :  S825  =  x  :  $33.  13.    §  :  f  r  i  64  :  x. 

7.  3  lb.  ;a;  =  §  yd.  :3yVyd.  14.    .36:  a;  : :  .125  :  4. 

8.  55.  77  =x:  42.  15.    $8  :  $2^  : :  $144  :«. 

16.  42  buo  s  36  bu.  : :  a;  :  3  pk. 

17.  184  mi.  ix  • :  75  mi.  :  525  ml. 

18.  X  :  320  A=  i  J  I  A.  :  g  A. 

19.  3,200  lb.  .  200  lb.  : :  96  lb.  :x. 

20.  468  rd.  :  920  rd.  : :  x  ;  2760  rd. 


PROPORTION.  30-9 

21.  i  :  H  : :  3  :  a:. 

22.  2^:7::  .0084  :  x. 

23.  63  gal.  :  90  gal.  ::x  :  S120. 
Note.     Is  the  above  a  true  proportion  ? 

24.  75  lb.  :  40  lb.  : :  60  d.  :  x. 

327.  1.  The  proportion  ma}'  be  used  in  the  solution  of 
problems  in  which  three  numbers  are  given  with  which  to 
find  a  fourth,  if  two  of  the  numbers  have  the  same  ratio  to 
each  other  that  the  third  has  to  the  required  number. 

2.  To  solve  such  a  jirobleni,  state  it  in  the  form  of  a 
proportion  in  which  the  required  number  is  the  fourth 
term,  and  the  tivo  given  related  numbers  are  the  first 
couplet,  determining  their  order  by  the  nature  of  tfit 
problem. 

PROBLEMS. 

1.  If  12  yards  of  cloth  cost  S36,  what  will  40  yards  cost 
at  the  same  price  ? 

Analysis.     12  yards  is  the  same  part  of  40  yards  that  the  cost  of  12 
yards  is  of  the  cost  of  40  yards ,  hence,  the  proportion  is  stated  as  follows ; 
1 2  yards  :  40  yards  : :  $36  :  x. 
To  find  the  4th  term,  apply  Principle  3. 

2.  If  50  bushels  of  coru  cost  S20,  what  will  600  bushels 
cost  at  the  same  price  ? 

3.  If  a  man  travel  48  miles  in  12  hours,  how  many  miles 
can  he  travel  in  60  hours  at  the  same  rate? 

4.  If  17  horses  cost  $1,360,  what  will  39  horses  cost  at 
the  same  price? 

5.  If  the  cost  of  1,360  square  yards  of  plastering  be 
S340,  what  will  3,824  yards  cost  at  the  same  price? 

6.  If  a  steeple  124  feet  high  cast  a  shadow  93  feet  long, 
how  long  a  shadow  will  a  steeple  216  feet  high  cast  at  the 

same  time  and  place? 
21A 


SIO  XEW   ADVAXCED  ARITHMETIC. 

7.  If  a  steeple  216  feet  high  cast  a  shadow  162  feet  long, 
how  long  a  shadow  will  be  cast  by  a  steeple  124  feet  high 
at  the  same  time  and  place  ? 

8.  Suppose  the  shadow  cast  by  a  216-foot  steeple  to  be 
162  feet  long,  what  is  the  length  of  a  steeple  that  casts  a 
93-foot  shadow,  the  other  conditions  being  equal  ? 

9.  If  a  124-foot  steeple  cast  a  93-foot  shadow,  what  is 
the  height  of  a  steeple  that  casts  a  162-foot  shadow,  under 
the  same  conditions? 

10.  If  25  men  can  do  a  piece  of  work  in  12  days,  in  how 
many  days  can  10  men  do  the  same  work? 

Note.  This  proportion  differs  from  those  preceding  it,  in  that  it  in- 
volves an  "inverse  proportion."  Since  the  length  of  time  wiU  increase 
•as  the  number  of  men  diminishes,  the  proportion  will  be  10  :  25  =  12  :  x. 
Be  ready  to  recognize  such  proportions. 

11.  If  10  men  can  do  a  piece  of  work  in  30  days,  how 
many  men  can  do  the  same  work  in  12  days? 

12.  If  a  principal  of  S32p  earn  876.50  in  a  given  time, 
-what  will  a  principal  of  8648  earn  in  the  same  time  ? 

13.  A  room  is  24  feet  wide  and  28  feet  long.  How  long 
must  a  room  18  feet  wide  be  to  contain  the  same  area? 

14.  If  50  acres  of  land  produce  2,800  bushels  of  corn, 
how  many  bushels  will  76  acres  yield  at  the  same  rate? 

15.  If  a  man  can  do  a  piece  of  work  in  36  days,  working 
10  hours  a  day,  in  how  many  days  could  he  do  the  same 
•working  12  hours  a  day? 

16.  A  railway  train  runs  429  miles  in  8  hours  and  15 
minutes.  How  far  would  it  run  in  10  hours  and  20  minutes 
at  the  same  rate? 

17.  If  3  lb.  5  oz.  of  butter  cost  81.06,  how  much  would 
8  lb.  12  oz.  cost  at  the  same  price? 

18.  A  garrison  of  320  men  is  supplied  with  provisions  for 
60  days.  How  long  would  the  remainder  of  the  stock  of 
supplies  last  the  rest  of  the  garrison,  if  40  men  were  with- 
drawn at  the  end  of  20  davs? 


I 


PROPORTION.  311 

19.  Two  cog  wheels  are  geared  together.  The  hirger  has 
42  cogs  and  the  smaller  16.  How  many  revolutions  does 
the  smaller  make  while  the  larger  makes'  21:  ? 

20.  If  120  bushels  of  oats  be  necessary  to  seed  40  acres 
of  land,  how  many  bushels  will  seed  195  acres? 

21.  If  '2%  barrels  of  pork  cost  823.625,  how  much  will 
8y^2  barrels  cost? 

22.  If  3i'o  acres  of  land  yield  172  bushels  of  wheat,  how 
much  will  18.75  acres  yield? 

23.  If  the  interest  on  S375.50  for  a  certain  time  is  852-57, 
what  is  the  interest  on  8680  for  the  same  time? 

328.    COMPOUND   RATIO. 

1.  The  value  of  a  ratio  is  expressed  by  making  the  ante- 
cedent the  numerator,  and  the  consequent  the  denominator, 
of  a  fraction. 

2.  Since  a  fraction  may  be  multiplied  by  a  fraction,  a 
ratio  may  be  multiplied  by  a  ratio. 

3.  The  ratio  of  two  numbers  is  called  a  Simple  Ratio. 

4.  The  indicated  product  of  two  or  more  simple  ratios  is 
a  Compound  Ratio. 

5.  Compound  ratios  are  usually  expressed  in  the  follow- 
ing form : 

6  :  10 

4  :    5 

2  :     7 

This  expression  is  read:  the  (Compound  ratio  of  6  to  10, 
4  to  5,  and  2  to  7.  Its  value  is  f^  of  f  of  2.  The  value  of 
a  compound  ratio  may  be  expressed  as  a  compound  fraction. 
Note.  Give  the  rule  for  multiplying  a  fraction  by  a  fraction.  Make 
a  rule  for  multiplying  a  ratio  by  a  ratio. 

Find  the  value  of  each  of  the  following  compound  ratios, 
employing  cancellation  where  possible  : 


312  NEW  ADVANCED  ARITHMETIC. 


8 


1. 


3  :5>        3     "  *  loC  _      7pt.  :  12pt.) 

4  :  7  r       ^-    r-^l\  ^-    4  A.  :  9  A.     ; 


5. 

7  pt.  : 
4  A.  : 

6. 

18  :  24 

30  :     9 

125 

:  600 

2.5 

:    75 

§ 

:      2 

n 

:  -10 

S  :  I  )  $3  :  $8  )  18  :  24  ) 

^  :  f  y  *    4  ft.  :  7  ft.  I  30  :     9  f 


7i  :  82  ) 
7.    15  :  2h  > 


329.  1.  A  Compound  Proportion  is  a  proportion  in  which 
there  is  a  compound  ratio. 

2.  The  following  problem  involves  a  compound  pro- 
portion : 

If  a  man,  working  8  hours  a  day,  build  60  feet  of  fence  in 
2  days,  how  many  feet  of  fence  can  a  man  build  in  6  days, 
working  10  hours  a  day? 

Note.     What  is  assumed  aboui  the  two  fences  ?     About  the  men  ? 

Analysis.  If  the  days  were  of  the  same  length,  tlie  proportion  would 
read,  2  days  :  6  days  : :  60  feet  :  x  feet.  Since  the  2  days  are  j%  as  long  as 
the  6  days,  the  woric  done  in  2  days  will  not  be  to  the  work  done  in  6  days 
as  2  is  to  6,  but  will  be  jg  of  that  ratio,  which  is  found  by  multiplying 
2  .  6  by  8  :  10. 

The  form  :  "  '    ^  i  .  GO  .  r 
8  .  10  ) 

Because  there  are  two  elements  involved  in  determining 
the  time  that  each  man  worked  (the  number  of  days  and 
their  length),  the  relation  of  the  times  is  expressed  by  a 
compound  ratio. 

RULE. 

IPo  solve  a  probletn  invotring  a  conrpound  if^oi>orfion, 
take  for  the  third  term  the  antecedent  of  the  ratio  of 
mhich  the  required  term  is  the  eonsequentm  With  one 
pair  of  the  remaining  terms  for  the  first  ratio,  stMe 
the  proportion  aeeording  to  the  con€litions  of  the  probtenu 

Proceed  with   the   feniaining  2>airs  in  the  same  way 


PROPORTION.  313 

until  all  of  the  conditions  are  stated.  Find  the  product 
of  the  third  term  and  all  of  the  second  terms,  and  divide 
it  by  the  product  of  the  first  terms, 

PROBLEMS. 

1.  If  4  men,  in  5  days  of  8  hours  each,  can  dig  a  ditch 
120  yards  long,  3  feet  wide,  and  4  feet  deep,  in  how  many 
days  of  10  hours  each  can  12  men  dig  a  ditch  300  yards 
long,  4  feet  wide,  and  4^  feet  deep? 

Statement :  120  yd.  :  30U  yd.] 

3  ft.  :  4  ft. 

4  ft.  :  4^  ft.  I  :  :  5  days  :  x 
12  men  :  4  men 
10  nr.  :  8  hr. 

Why  is  12  put  first  in  the  fourth  ratio?  Why  10  in  the 
last?     What  cancellation  can  be  employed?     Why? 

2.  If  S180  be  paid  for  the  work  of  5  men  for  24  days, 
what  should  be  paid  for  the  work  of  17  men  for  36  days? 

3.  If  15  men,  in  16  days  of  9  hours  each,  can  do  a  piece 
of  work,  how  many  men  will  be  needed  to  do  the  same  piece 
of  work  in  8  days  of  6  hours  each? 

4.  If  15  men,  in  16  days  of  9  hours  each,  can  do  a  certain 
piece  of  work,  in  how  many  days  of  6  hours  each  can  45 
men  do  the  same  work  ? 

5.  If  45  men  can  do  a  piece  of  work  in  8  days  of  6  hours 
each,  how  many  hours  a  day  must  15  men  work  to  do  the 
same  work  in  16  days? 

6.  If  50  tons  of  coal  are  required  to  run  4  engines  15 
hours  a  day  for  6  days,  how  many  tons  will  be  required  to 
run  7  engines  18  hours  a  day  for  11  days,  with  3  times  as 
heavy  a  load  ? 

7.  If  it  cost  $50  to  make  a  walk  8  feet  wide  and  60  feet 
long,  what  will  it  cost  to  build  a  walk  7|  feet  wide  and  72 
feet  long  ? 


514  NEW  ADVAXCED  ARITHMETIC. 

8.  If  it  cost  $50  to  make  a  walk  8  feet  wide  and  60  feet 
loug,  what  is  the  width  of  a  walk  that  is  72  feet  long,  and 
costing  $57.50? 

9.  If  a  walk  that  is  72  feet  long  and  7§  feet  wide  cost 
$57.50,  how  long  a  walk  that  is  8  feet  wide  can  be  built 
for  $50? 

10.  If  83  horses  eat  933  bushels  3  pecks  of  oats  in  30 
days,  how  many  bushels  will  125  horses  eal:  in  45  days? 

11.  If  44,640  bricks,  4  inches  by  8  inches,  will  pave  a 
court-yard,  how  many  tiles  8  inches  wide  and  15  inches  loug 
will  pave  it? 

Note.  Solve  these  problems  by  straight-line  analysis.  Additional 
problems  may  be  formed  from  those  given,  as  illustrated  in  7  and  8. 

Remark.  No  problem  can  be  solved  by  proportion  that 
cannot  be  more  easily  solved  by  straight-line  analysis. 

330.     PARTNERSHIP. 

1.  A  and  B  went  into  business  together,  A  investing 
$5,000,  and  B  $7,000.  They  agreed  to  share  gains  and 
losses  in  proportion  to  their  investments.  The  net  gain  was 
$2400.     What  was  each  one'o  share? 

(a)  Solve  by  analysis. 

What  was  the  whole  investment?  What  part  of  it  did 
A  invest?  What  part  of  it  did  B  invest?  What  was  A's 
share?     B's? 

(6)  Solve  by  proportion. 

Explain  these  proportions. 

(1)  $12000  :  85000  : :  $2400  :  A's  share. 

(2)  $12000  :  $7000  : :  $2400  :  B's  share. 

2.  The  action  of  A  and  B  is  called  "the  formation  of  a 
partnership."  The  amount  invested  is  called  the  Capital. 
A  and  B  are  called  Partners.  The  agreement  into  which 
they  enter  is  called  the  Conditions  of  Partnership. 

3.  A  Partnership  is  an  association  of  persons  for  the 
prosecution  of  business  on  joint  account. 


PROPORTION.  315 

PROBLEMS, 

1.  A.  B,  and  C  formed  a  two-year  partnership,  agreeing 
to  share  gains  and  losses  in  proportion  to  their  investments. 
A  put  in  $5,000 ;  B,  $6,000 ;  C,  $7,000.  Their  net  gain 
was  $8,000.  Find  the  share  of  each  by  analysis  and  by 
proportion. 

2.  If  their  net  loss  had  been  $2,1  GO,  what  would  have 
been  the  loss  of  each? 

3.  A,  B,  and  C  formed  a  partnership,  C  being  a  silent 
partner.  A  invested  $6,000;  B,  $8,500;  C,  $10,000.  By 
the  conditions  of  partnership,  A  was  to  receive  a  salary  of 
$1,000,  and  B,  $700.  The  net  profits  were  to  be  divided  in 
proportion  to  investments.  At  the  end  of  the  first  year  the 
profits,  exclusive  of  all  expenses  but  salaries,  were  $4,150. 
What  was  the  share  of  each? 

4.  A,  B,  and  C  formed  a  partnership  for  3  years.  They 
were  to  draw  equal  amounts  as  salaries,  and  were  to  share 
the  net  profits  equitably.  A  invested  $5,000  at  the  begin- 
ning, added  $3,000  to  it  at  the  end  of  the  first  year,  and 
$2,000  more  at  the  end  of  tlie  second.  B  mvested  $6,500  at 
the  beginning,  withdrew  $2,000  at  the  end  of  the  first  year, 
and  $1,500  at  the  end  of  the  second.  C  invested  $5,000  at 
the  beginning,  and  did  not  change  it.  At  the  end  of  the 
time  their  net  profits  were  $4,850.  What  was  the  share 
of  each? 

5.  The  investments  of  three  partners  are  in  the  ratio  of 
3,  4,  and  5.    If  they  gain  $3,600,  what  is  the  share  of  each? 

6.  A,  B,  and  C  owned  a  mill  valued  at  $18,000.  A  owned 
J  of  it :  B,  I  of  it ;  and  C,  the  rest.  It  was  insured  for  f  of 
its  value.  If  it  should  be  destroyed  by  fire  what  would  each 
partner  lose? 

7.  A,  B,  C,  and  D  constructed  a  street  railroad  costing 
$135,000.  A  furnished  J  of  the  capital;  B,  ^  of  it;  rnd  C 
and  D  each  furnished  ^  of  the  remainder.     The  company 


/ 


316  NEW  ADVANCED  ARITHMETIC. 

sold  to  E  t's  of  the  road  for  S9,000  ;  what  part  of  this  amount 
should  each  receive  ?  What  part  of  the  stock  would  each  of 
the  original  partners  own  after  the  sale? 

8.  A,  B,  and  C  tuok  a  contract  for  excavating  a  railroad 
cut.  A  furnished  60  men  for  25  days;  B,  50  men  for  48 
days;  C,  75  men  for  56  days.  They  received  ^20,250  for 
the  work.     What  was  the  share  of  each? 

9.  A,  B,  and  C  engaged  in  business  for  one  year,  agreeing 
to  share  the  profits  in  proportion  to  their  investments.  On 
January  1,  A  put  in  $3,000  ;  B,  $3,500;  and  C,  82,500.  On 
March  1,  A  increased  his  share  $500,  B  diminished  his  $500, 
and  C  increased  his  $250.  On  July  1,  A  withdrew  $1,000, 
B  put  in  $800,  and  C  increased  his  $1,000.  On  October  1, 
A  put  in  $600,  B  withdrew  $400,  and  C  withdrew  $750. 
They  gained  $3,000.     What  was  the  share  of  each? 

10.  A,  B,  and  C  engaged  in  business  for  2  years,  with  a 
capital  of  $16,000.  A  furnished  a  and  B  -^  of  the  capital. 
C  conducted  the  business  for  one  half  the  net  profits.  The 
gross  earnings  were  $4,800.  The  expenses  were  12i%. 
What  was  A's  share?   B's? 

11.  What  would  have  been  A's  share,  if  at  the  end  of  the 
first  year  he  had  transferred  to  B  one  thu'd  of  his  interest? 


INVOLUTION.  317 

SECTION     X. 

331.     INVOLUTION. 

1.  Multiply  each  of  the  following  numbers  by  itself:  7,  9, 
13,  24,  48,  69.     The  result  in  each  of  these  cases  is  a  Square. 

2.  The  Square  of  a  number  is  the  product  arising  from 
multiplying  that  number  by  itself.  The  result  is  also  called 
the  Second  Power  of  the  number. 

3.  Learn  the  squares  of  all  whole  numbers  from  1  to  25. 

4.  The  expression  8-  indicates  that  8  is  to  be  used  twice 
as  a  factor.  8''^  =  8  X  8  =  64.  The  2  as  here  used  is  called 
aai  Exponent. 

5.  An  Exponent  is  an  expression  placed  at  the  right  of 
and  above  a  number  to  indicate  how  many  times  it  is  to  be 
used  as  a  factor. 

6.  12^  =  ?     18^=?     1252=?     (|)2=r?     21-=?     .0152=? 

7.  3x3x3=?  This  result  is  the  cube,  or  thii'd  power, 
of  3.     What  is  the  cube  of  4  ?  of  5  ?  of  8  ?  of  10  ? 

8.  The  Cube  of  a  number  is  the  product  arising  from  using 
that  number  three  times  as  a  factor. 

9.  Make  a  definition  for  the  Fourth  Power  of  a  number; 
for  the  Fifth  Power ;  for  the  Eighth  Power. 

10.  53  =  ?  83  =?  (3)3=?  73=?  2*=?  3^  =  ? 
a)*  =  ?     (§)'  =  ?     -05"  =  ? 

11.  25'-=?     16^  =  ?     (^1)3  =  ?     32*=?     2.5*=?     .02^=? 

12.  Learn  the  cubes  of  all  integers  from  1  to  10, 

13.  Recite  the  following  rapidly:  19^,  W,  \2\  b\  U\  7\ 
222;  43^  18S  2P,  103,  g3^  232,  132,  83,  172,  15^  93,  20S  14^,  25*. 


318  NEW  ADVAXCED   ARITHMETIC 


333.     EVOLUTION. 

1.  Name  one  of  the  two  equal  factors  whose  product  is 
4,  9,  25,  49,  81,  169. 

2.  Name  one  of  the  three  equal  factors  whose  product  is 
8,  27,  64,  125,  343,  729. 

3.  Name  one  of  the  four  equal  factors  whose  product  is 
16,  81,  256,  625. 

4.  Each  of  the  preceding  results  is  a  Roo.t. 

5.  A  Root  of  a  number  is  one  of  the  equal  factors  whose 
product  is  the  number. 

6.  The  Square  Root  of  a  number  is  one  of  the  two  equal 
factors  whose  product  is  the  number. 

7.  Define  the  Cube  Root  of  a  number;  the  Fourth  Root; 
the  Sixth  Root. 

8.  What  is  the  square  root  of  49?  of  81  ?  of  144?  of  324? 
of  441?  of  625.^ 

9.  Evolution  is  the  process  of  finding  the  root  of  a 
number. 

10  ^4  =  2.  ^25  =  5.  \/iU  =  12.  The  sign  placed 
before  these  numbers  is  called  the  Radical  Sign.  It  indicates 
that  a  root  of  the  number  is  to  be  found. 

11.  ^8  =  2.  ^64  =  4.  ^81-3.  ^32  =  •-?•  The 
expression  placed  above  the  sign  is  called  the  Index.  The 
radical  sign,  when  used  alone  before  a  number,  indicates  that 
its  square  root  is  to  be  extracted. 

12  The  index  is  read  as  its  corresponding  ordinal  number. 
^8  is  read,  "  the  third  root  of  8,"  or  "  the  cube  root  of  8." 

Read  ^fe;  -^32;  Vh  VW- 

13.  ^324,  a/^,  \/T2],  V^,  \/r2T,  VI^ST,  V-0016. 

14.  Instead  of  the  radical  sign,  a  fractional  exponent  may 
be  employed.     9=  =  \/9.     27^  =  -^27^     64^  =  ^64^ 


EVOLUTION.  319 

15.  To  understand  how  numbers  may  be  separated  into 
equal  factors,  let  us  study  their  composition. 

16.  Since  the  square  of  1  is  1,  and  of  9  is  81,  it  is  clear 
that  the  square  of  a  one-place  number  cannot  be  more  than  a 
two-place  number.  The  square  of  10,  the  smallest  two-place 
number,  is  100,  a  three-place  number.  The  square  of  99, 
the  largest  two-place  number,  is  9,801,  a  four- place  number. 

17.  Similarly  it  may  be  shown  that  the  square  of  any  in- 
tegral number  contains  twice  as  many  places  as  the  number, 
or  one  less  than  twice  as  many.  Conversely,  if  any  integral 
number  be  separated  into  periods  of  two  places  each,  begin- 
ning with  units,  the  number  of  periods  thus  formed  will  be 
the  same  as  the  number  of  places  in  the  square  root.  The 
left-hand  period  may  have  but  one  place. 

18.  What  is  the  square  of.l  ?  of  .9  ?  of  .01  ?  of  .09  ?  The 
square  of  any  decimal  fraction  will  contain  how  many  times 
as  many  decimal  places  as  the  fraction?     Why? 

19.  To  find  the  number  of  places  in  the  square  root  of  a 
decimal  fraction : 

a.  If  the  fraction  has  an  odd  number  of  places,  annex 
one  zero.     Why? 

h.  Beginning  with  tenths'  order,  separate  the  expression 
at  the  right  of  the  decimal  point  mto  periods  of  two  places 
each. 

20.  Study  the  form  of  the  square  of  64. 


X 

4 
6  +  6 

64 
64 

6 

X 
X 

6  +  4x4 
4 

6 

X 

^  +  ■2 

X 

4  X  6  +  4  x 

4 

It  is  clear  that  the  square  of  any  two-place  number  must 
take  this  form:  The  square  of  the  tens  +  twice  the  product 
of  the  tens  and  units  +  the  square  of  the  units.     Remem- 


320  NEW  ADVANCED  ARITHMETIC. 

bering  this  fact,  the  square  root  of  a  four-place  number  may 
be  found  as  follows : 


Illustrative  Problem.     21.    V'44:89  =  ? 

Since  4489  is  a  four-place  number,  its  square  root  must  be 
a  two-place  number.  Siuce  the  square  of  tens  is  hundreds, 
it  must  be  fouud  in  44  hundreds.  The  largest  square  in  44 
hundreds  is  36  hundreds.  Its  square  root  is  6  tens.  Sub- 
tracting 36  hundreds,  the  remainder  is  8  hundreds.  The 
entire  remainder  is  889.  If  the  original  number  is  a  square, 
889  is  the  sum  of  twice  the  tens  of  the  root  by  the  units,  and 
the  square  of  the  units.  The  tens'  term  is  6.  Twice  the 
tens  is  12  tens.  Tl:e  product  of  the  12  tens  and  the  units  is 
tens ;  hence,  it  must  be  in  88  tens.  7  is  probably  the  units' 
term  of  the  root.  7  X  12  tens  is  84  tens.  88  tens  —  84 
tens  =:  4  tens.  4  tens  +  9  =  49.  49  is  the  square  of  7; 
hence,  the  square  root  of  4489  is  67. 

Note.  If  the  number  should  be  a  pure  or  mixed  decimal,  it  may  be 
considered  an  integer,  and  the  result  mny  be  corrected. 

FORM. 

4489  I  67 
36 
1^  1^88 
84 

49 

Note.  Explain  the  following  form,  showing  that  it  is  briefer  than  the 
{oTjuer. 

4489   |"67 
36 


127  I     889 
889 

22.    How  can  this  plan  be  extended  to  larger  numbers? 
Illustrative  Problem.     V288369  =--  ? 


EVOLUTION.  321 

By  examining  the  number  it  is  found  that  its  root  is  a 
three-place  number.     We  may  first  deal  with  2883. 

FORM. 

2883  [-53 
25 
103  1^83 
309 

53  may  now  be  regarded  as  the  tens*  teiJQ  of  the  root. 
To  the  remainder,  74,  the  remainder  69  may  be  annexed, 
and  the  work  continued  as  before. 

FORM. 

288369  I  537 
25 


103  |~^^ 
309 

1067  I       7469 

333.     RULE   FOR  THE   EXTRACTION   OF   THE   SQUARE    ROOT 
OF  A   NUMBER. 

1.  Beginning  at  the  decimal  point,  group  the  figures 
into  periods  of  ttvo  orders  each. 

2.  Find  the  largest  square  in  the  left-hand  period  and 
place  its  root  at  the  right,  as  the  first  term  of  the  root. 

3.  Subtract  the  square  from  the  left-hand  period,  and 
to  the  remainder  annex  the   next  period. 

4.  Double  the  root  already  found,  and  using  it  as  a 
trial  divisor  find  how  many  times  it  is  contained  in  the 
new  dividend  exclusive  of  its  right-hand  term.  Place  the 
quotient  as  the  second  terin  of  the  root,  and  also  annex  it 
to  the  trial  divisor.  Multiply  the  complete  divisor  by  the 
second  term  of  the  root,  subtract  the  product  from  the 
partial  dividend,  and  proceed  as  before. 


322 


NEW  ADVANCED  ARITHMETIC. 


5.  If  the  trial  dividend  tvill  not  contain  the  trial  dit'isorf 
annex  a  sero  to  the  root  and  to  the  trial  divisor,  annex  a 
new  period  to  the  trial  dividend,  and  proceed  as  before. 

6.  If  there  is  a  remainder  after  the  last  operation,  to 
continue  the  tvork  reduce  the  remainder  to  hundredths, 
ten-thousandths,  etc.,  continuing  the  tvork  as  before. 

7.  To  extract  the  square  root  of  a  common  fraction 
where  the  terms  are  squares,  extract  tJie  square  root  of 
each  term.  If  only  the  denominator  is  a  square,  extract 
the  approximate  root  of  the  numerator  and  divide  it  by 
the  root  of  the  denominator. 

If  the  denominator  is  not  a  square,  change  the  fraction 
to  a  decimal  and  extract  its  approximate  root. 

334.  1.  Since  the  number  of  units  in  the  area  of  a 
square  is  the  square  of  the  number  of  units  in  one  side, 
the  square  root  of  the  number  of  units  of  area  is  the  num- 
ber of  units  in  a  side ;  hence,  if  the  area  of  a  square  be 
given,  a  side  may  be  f  Dund  by  applying  the  preceding  rule. 
The  process  of  applying  the  rule  may  also  be  illustrated  by 
diagrams. 

2.  What  is  one  side  of  a  square  whose  area  is  1225  square 
feet? 

First,  confining  our  attention  to  the  left-hand  period,  12, 
it  appears  that  this  square  contains  at  least  900  square  feet. 
Each  side  of  such  a  square  is  30  feet. 

E  F  M 


A 

30  ft. 

J 

900  sq.  ft. 

i 

0 

c 

H 


EVOLUTION.  323 

There  yet  remain  325  square  feet  with  which  to  make 
additions  to  this  figure.  They  must  be  made  so  as  to  i-etain 
the  form  of  a  square  ;  hence,  the  length  and  width  must  be 
equally  increased.  If  a  rectangle  a  foot  wide  were  to  be 
added  to  the  sides  A  B  and  B  C,  2  X  30  square  feet  (twice 
the  tens)  would  be  needed.  The  remainder  is  large  enough 
to  make  these  additions  5  feet  wide,  and  leave  25  square  feet 
with  which  to  fill  out  the  space  F  M  G  B. 

Note.  Use  the  figure  to  illustrate  the  extraction  of  the  square  root  of 
541696. 

PROBLEMS. 
Find  the  square  root  of  each  of  the  foiiowing  numbers: 


1.  4489. 

8. 

143641. 

15. 

40640625. 

2.  7921. 

9 

214369. 

16. 

9036036. 

3.  9216. 

10. 

450241. 

17. 

23261329. 

4.  15625. 

11. 

466489. 

18. 

.6889. 

5.  42436. 

12. 

519841. 

19. 

.355216. 

6.  82369. 

13. 

567009. 

20. 

76.3876. 

7.  93025. 

14. 

622521. 

21. 

49 
64* 

Find  the  root  to  hundredths. 

22.  if. 

26. 

28. 

30. 

3.6. 

23.  \n- 

27. 

6. 

31. 

4.9. 

24.  l%%- 

28. 

2. 

32. 

8.1. 

25.  m- 

29. 

.7. 

33. 

12.1. 

See  Art.  333,  7. 

34.  A|. 

38 

§• 

42. 

121. 

35.  fs. 

39 

7 

43. 

81. 

36.  I. 

40. 

H- 

44. 

^. 

37.  it\. 

41. 

n- 

45. 

161. 

335.     1.    Find  the  length  of  one  side  of   a  square  field 
containing  17  A.  89  sq.  rd. 


324 


NEW  ADVANCED  ARITHMETIC. 


2.  The  entire  surface  of  a  cubical  block  contains  223^5 
square  feet.     "Wliat  is  the  length  of  one  edge? 

3.  A  square  contains  900  square  inches.  What  are  the 
width  and  length  of  an  equivalent  rectangle  whose  width  is 
to  its  length  as  1  to  4  ? 

4.  Supply  a  mean  proportional  in  each  of  the  following 
proportions : 

a.     I2ix':ix:4:8.  c.    4.8:x:tx.  432 

6.  192:  a;::  a;:  27. 

KoTE.  A  mean  proportional  is  a  number  that  is  the  second  and  third 
term  of  a  proportion. 

5.  How  many  rods  of  fence  will  enclose  a  square  field 
containing  10  acres? 

60  A  body  of  7,921  soldiers  is  arranged  so  that  there  are 
as  many  in  rank  as  there  are  in  file.  How  many  are  there  in 
each  ? 

7.  A  rectangular  surface  whose  width  is  |  ot  its  length 
contains  1^470  square  feet     Find  its  width  and  length 

336      THE   RIGHT 
TRIANGLE 

1.  A  triangle,  one 
of  whose  angles  is  a 
right  angle,  is  called  a 
right  triangle.  The 
side  opposite  the  right 
angle  is  the  hypote- 
nuse. The  other  sides 
are  the  base  and  alti- 
tude The  sides  form- 
ing the  right  angle  are 
called  the  arms. 

2.  Draw  a  right  tri- 
angle, ABC,  on  a 
sheet   of    pasteboard. 


EVOLUTION.  325 

Draw  the  squares  on  the  three  sides  and  subdivide  as  shown 
in  this  figure,  by  extending  the  sides  of  the  largest  square 
through  the  smaller  squares,  and  drawing  a  line  at  right 
angles  to  the  longer  extension.  With  a  sharp  knife  cut 
out  the  five  pieces  and  place  them  in  the  positions  1',  2',  3', 
etc.  We  thus  see  that  the  square  of  the  hypotenuse  is  equal 
to  the  sum  of  the  squares  of  the  arms. 

PROBLEMS.  ,  " 

Base.  Altitude.  Hypotenuse. 

1.  8  inches.  ?  10  inches. 

2.  20  feet.  15  feet.  ? 

3.  224  yards.  ?  260  yards. 

4.  ?  272  mUes.  353  miles. 

5.  192  rods.  144  rods.  ? 

6.  The  top  of  a  ladder  that  is  30  feet  long  rests  against  n 
telegraph  pole  24  feet  from  the  ground ;  how  far  is  the  foot 

of  the  ladder  from  the  foot  of  the  pole  ?  >^ 

7.  A  and  B  start  from  the  same  point  at  the  same  time.     ^ 
A  travels  north  and  B  east,  the  former  traveling  at  the  rate 

of  four  miles  an  hour  and  the  latter  three.  How  many  feet 
apart  are  they  in  15  minutes? 

8.  A  rope  is  attached  to  the  top  of  a  96-foot  pole.  It 
touches  the  ground  28  feet  from  the  foot  of  the  pole.  What 
is  its  length  ? 

9.  What  is  the  diagonal  of  a  rectangle  whose  dimensions 
are  6  yards  and  8  yards? 

Note.  Tlie  diagonal  of  a  rectangle  is  the  straight  line  joining  opposite 
vertices. 

10.  A  13-foot  ladder  rests  with  its  top  against  a  window 
sill  12  feet  from  the  ground.  How  many  feet  from  the  wall 
to  the  foot  of  the  ladder? 

11.  Against  the  top  of  a  pole  15  feet  high  are  braced  in 
opposite  [directions  a  17-foot  ladder  and  a  25-foot  ladder. 
How  far  apart  are  the  feet  of  the  ladders.     Make  diagram. 

22A 


326 


NEW  ADVANCED  ARITHMETIC. 


12.  A  25-foot  ladder  stands  erect  against  the  wall  of  a 
building.  The  foot  is  pulled  out  until  the  top  is  lowered  one 
foot.     How  far  from  foot  of  ladder  to  wall  ? 

13.  If  the  foot  of  the  ladder  is  pulled  out  8  feet  farther, 
how  much  more  is  the  top  lowered? 

14.  What  is  the  area  of  a  rectangle  whose  length  and 
diagonal  are  respectively  15  and  17  rods? 

15.  What  is  the  length  of  the  longest  wire  that  can  be 
stretched  straight  in  a  room  20'  X  15'  X  12'? 

16.  What  is  the  diagonal  of  an  inch  cube? 

17.  What  is  the  shortest  line  that  can  be  traced  on  the 
surface  of  a  cube,  joining  the  extremities  of  a  diagonal? 

18.  What  is  the  diameter  of  a  circle  that  will  just  enclose 
four  silver  dollars  arranged  in  the  form  of  a  square  ? 

19.  How  much  is  saved  by  going  diagonally  across  a  sec- 
tion instead  of  along  the  boundary? 

20.  How  many  acres  in  a  square  field  whose  diagonal 
exceeds  its  side  by  16.568  rods? 

Note.     The  valiie.s  of   V2  aud   V'3  should  be  memorizeiL 

21.  A  rectangular  field  three  times  as  long  as  wide  con- 
tains 87  acres.     What  is  its  length? 

22.  A  triangular  field,  A  B  C,  is  60  rods  long  and  20  rods 
wide.  At  what  distance  from  B  may  a  fence  be  built  divid- 
ing the  field  into  two  equal  areas? 

B 


EVOLUTION.  327 

Queries.  A  C  is  what  part  of  A  B  ■?  X  O  is  what  part  of  X  B' 
What  is  the  area  of  rectangle  O  X  B  T  1  What  is  the  area  of  tlie  square 
O  X  K  S  ?     What  is  the  length  of  O  X  ?  of  X  B  ^ 

23*  At  what  point  shall  a  triangular  board,  12  feet  long 
and  12  inches  wide,  be  sawn  into  two  equivalent  pieces? 

24.  At  what  point,  if  the  12-foot  board  be  a  trapezoid,  15 
inches  wide  at  one  end,  3  inches  at  the  other  ? 

25.  Show  that  128  stakes  a  foot  apart  can  be  driven  on  a 
ten-foot  square. 

26.  The  sides  of  an  equilateral  triangle  are  six  feet  each. 
What  is  the  altitude  ? 

27.  What  is  the  area  of  an  equilateral  triangle,  each  side 
of  which  is  4  feet  ? 

28.  Draw  the  three  altitudes  of  an  equilateral  triangle  from 
the  vertices  to  the  opposite  sides.  They  meet  at  a  common 
point.  Join  this  point  with  the  vertices  forming  three  equal 
triangles.  Show  that  the  altitude  of  each  is  i  the  altitude  of 
the  equilateral  triangle. 

29.  What  is  the  diameter  of  the  circle  that  will  just  enclose 
three  silver  dollars  arranged  in  the  form  of  a  triangle? 

30.  What  is  the  diameter  of  a  circle  whose  area  is  314.16 
square  feet? 

31.  With  what  length  of  rope  shall  a  horse  be  tethered  to 
graze  over  one  fourth  of  an  acre? 

32.  A  road  two  rods  wide  about  a  square  field  contains 
one  acre.     What  is  the  area  of  the  field  ? 

33.  A  road  two  rods  wide  about  a  circular  field  contains 
one  acre.     AVhat  is  the  area  of  the  circle? 

34.  About  a  circular  field  80  rods  in  diameter  is  a  road  of 
uniform  width  containing  six  acres.     Width  of  the  road? 

35.  How  much  must  the  diameter  of  a  36-inch  grindstone 
be  reduced,  to  reduce  the  weight  one  fourth?  (No  allowance 
for  the  opening.) 

*  A  riglit  triangle.  Use  figure,  page  326.  A  C  is  ^^  of  A  B.  Assume 
X  dividing  point.  X  O  is  j^j  of  X  B.  Divide  X  B  into  12  parts.  Area  of 
\'Rn'2     Of  XBTO?      Of  each  of  the  12  squares?     Length  of  X  0  ? 


328  NEW  ADVANCED  ARITHMETIC. 


337.     CUBE   ROOTc 

1.  What  is  5  X  5  X  5  ?  9x9x9?  23  X  23  X  23  ? 
86  X  86  X  86  ?  2  X  §  X  §  ?  i  X  i  X  i  ?  .8  X  .8  X  .8  ? 
.04  X  .04  X  .04?  The  product  arising  from  each  of  these 
indicated  multiplications  is  a  Cube. 

2.  The  cube  of  a  number  is  the  product  arising  from 
using  that  number  three  times  as  a  factor. 

3.  What  is  the  cube  of  79  ?  93?  207?  300?  |?  -j'^? 
U?     1.6?     .24?     10^=?     f5=?     (})'  =  ?     .06^  =  ? 

4.  What  is  one  of  the  three  equal  factors  whose  product 
is  8?     27?     125?     729?    ^y^?     ^?     .216?     .000512?    3a? 

5.  The  Cube  Root  of  a  number  is  one  of  the  three  equal 
factors  whose  product  is  the  number. 

6.  4^8='}     'V^27=?     \^Wi='^     \/:729=? 

7.  Define  Exponent,  Radical  Sign,  Index,  Fractional 
Exponent. 

338.     EXTRACTION  OF  THE  CUBE  ROOT  OF  NUMBERS. 

1.  The  method  of  extracting  the  cube  root  of  any  number 
above  1000  will  be  ascertained  by  studying  the  form  of  the 
cube  of  a  two-place  numbero 

2.  46^  =  46  X  46  X  46  =  (4  ^  +  6)  (4  ^  +  6)  (4  ^  +  6). 


1 
+  2 

« 

(4  0== 

4^+6 
4^+6 
4  <  X  6  +  62 
+  At  X  Q 

(4  0^+2 

X  4  f  X  6  +  6- 

4<  +  6 

(4  0« 

X 
X 

(4  0^ 

(4  0' 

X  6  +  2 
X  6  +  1 

X  4 «  X  6-  +  6^ 
X  4  «  X  62 

C4  0'  +  3  X  (4  0*^  X  6  +  3  X  4 «  X  6-  +  6^ 


EVOLUTION.  329 

3.  Stating  the  above  result  in  words,  we  have  the 
following : 

The  cube  of  a  t-wo-place  number  consists  of  (1)  the  cube 
of  the  tens,  (2)  plus  three  times  the  square  of  the  tens  by  the 
units,  (3)  plus  three  times  the  tens  by  the  square  of  the  units, 
(4)  plus  the  cube  of  the  units. 

Note.     Verify  the  above  statement  aud  fix  it  iu  the  memory. 

4.  Illustrative  Ex-ample.,  The  cube  of  35  =  SO**  or  27000' 
+  3  X  30-  X  5,  or  13500  +  3  X  30  X  5",  or  2250,  +  5%  or  125. 

Give  the  several  parts  of  the  cube  of  24 ;  of  32  ;  of  41 ; 
of  66;  of  87;  of  93. 

5.  The  cube  of  1  is  1.  The  cube  of  9  is  729.  Therefore, 
the  cube  of  a  one-place  number  is  not  more  than  a  three- 
place  number. 

6.  The  cube  of  10  is  1000.  The  cube  of  99  is  970299. 
Therefore,  the  cube  of  a  two-place  number  cannot  be  less- 
than  a  four-place,  nor  more  than  a  six-place,  number. 

7.  Similarly,  it  may  be  shown  that  the  cube  of  any  in- 
tegral number  contains  three  times  as  many  orders  as  the 
number,  or  three  times  as  many  less  one  or  two. 

8.  The  number  of  orders  in  the  root  of  an  integer  may  be 
ascertained  by  beginning  at  the  decimal  point,  and,  so  far 
as  possible,  grouping  the  figures  into  periods  of  three  orders 
each.  The  left-hand  period  may  contain  only  one  or  two 
figures. 

339.    Illustrative  Example.     ^^103823  =? 

Since  this  is  a  six-place  number,  the  cube  root  of  the 
largest  cube  in  it  is  a  two-place  number.  The  root,  there- 
fore, consists  of  some  number  of  tens  plus  some  number  of 
units. 

I  first  withdraw  from  the  number  the  largest  cube  in  103 
thousands.  This  is  64  thousands.  Its  cube  root  is  4  tens. 
103823  —  64000  =  39823,  We  have  seen  that  the  second 
part  of  the  cube  is  "three  times  the  square  of  the  tens  by 


330  NEW  ADVANCED  ARITHMETIC. 

the  units."  Three  times  the  square  of  the  tens  is  48  hun- 
dreds. If  this  were  multiplied  by  the  units  of  the  root,  the 
product  would  be  hundreds ;  hence,  would  not  be  of  a  lower 
denomination  than  398  hundreds.  The  398  hundreds,  then, 
may  be  used  as  a  trial  dividend  with  which  to  find  the  units' 
term  of  the  root.  The  trial  dividend  seems  large  enough  to 
indicate  that  the  units'  term  is  8,  but  it  is  to  be  remembered 
that  there  are  two  more  parts  to  the  cube.  Trying  7  as 
the  units'  term,  "three  times  the  square  of  the  tens  by  the 
units"  is  336  hundreds.  "Three  times  the  tens  by  the 
square  of  the  units"  is  588  tens.  "The  cube  of  the  units" 
is  343.  The  sum  of  these  numbers  is  39823;  hence,  103823 
is  a  cube,  and  its  root  is  47. 


3  X  (4  fi)  =  48  hundreds 


103823  1 47 
_64 

330  =  7  X  48  hundreds. 
622 
588  =  3  X  4  i  X  72 

343 

343  =  73 


Since  "three  times  the  square  of  the  tens"  is  to  be  mul- 
tiplied by  the  units,  and  since  "  three  times  the  tens  by  the 
square  of  the  units "  may  be  found  by  multiplying  three 
times  the  tens  by  the  units,  and  that  product  by  the  units., 
and  since  "  the  cube  of  the  units  "  equals  the  square  of  the 
Mnits  by  the  units,  it  is  clear  that  the  units'  term  is  a  factor 
©f  each  of  these  parts  of  the  cube.  The  three  parts  may  be 
reduced  to  one  by  arranging  them  as  follows:  (-^  f^  +  3fn 
+  w'^)  n.  Here  3  t'^  is  the  trial  divisor,  and  (3  i^  +  3  tu  +  u'-) 
is  the  complete  divisor. 


EVOLUTION.  331 

FORM. 

103823  {_47 
G4 
3  X  /2  =  4800 
3  <  u  =  840 
u-   =   49 

5689  39823 

Note.     3  f  u  is  one  order  lower  tlian  3  /-.    Why  1    u-  is  one  order  lower 
than  3i  u.     Why  i     Why  are  the  two  zeros  placed  at  the  right  oi  48  ? 


39823 


PROB 

LEMS. 
6. 
7. 
8. 
9. 
10. 

1. 

^^79507  =  ? 

\  6585U3  = 
'^  804357  = 

? 

2. 

'^  157464  =  ? 

V 

3. 

-^314432  =  ? 

'V^884736  = 
-^970299  = 

? 

4. 

-v/357911  =  ? 

y 

5. 

V  551368  =  ? 

■V'912673  = 

V 

Note.  Pupils  should  work  on  similar  problems  until  they  can  be 
solved  easily  at  the  rate  of  one  a  minute. 

340.     DECIMAL    FRACTIONS. 

1.  The  cube  of  .1  is  .001.  The  cube  of  .9  is  .729.  The 
cube  of  .01  is  .000001.  The  cube  of  .09  is  .000729.  The 
cube  of  tenths  is  thousandths ;  of  hundredths  is  millionths ; 
of  thousandths  is  billionths,  etc. 

2.  To  find  the  number  of  orders  in  the  cube  root  of  a 
decimal  fraction,  begin  at  the  decimal  point  and  group 
the  figures  into  periods  of  three  orders  each. 

If    the    right-hand    jteriod    is    not    full,    annex    zeros. 

(Why?) 

PROBLEMS. 

Proceed  as  with  whole  numbers. 

1.  -^.032768  =  ?         3.    '^. 000195112  =  ? 

2.  -v^.07950T  =  ?         4.    v^. 000000493039  =  ? 


332 


NEW  ADVANCED  ARITHMETIC. 


341.     COMMON   FRACTIONS. 

1.  How  is  the  cube  of  a  common  fraction  found?     How, 
then,  may  the  cube  root  of  a  common  fraction  be  found? 

V27    —   •  V512    —   •  *'T"5  74g4    —    . 

Make  a  rule  for  the  extraction  of  the  cube  root  of  a  com- 
mon fraction. 

2.  If  the  denominator  is  not  a  cube,  change  the  common 
fraction  to  a  decimal  fraction,  and  proceed  as  in  Art.  340. 

342.    '^I0l847563  =  ? 


OPERATION. 


4800 

720 

36 

101847563 
64 

3  X42  = 

3X4x6  = 

6X6  = 

37847 

5556 

33336 

3  X  46-^  = 

3  X  40  X  7  = 

7X7  = 

634S00 

9660 

49 

4511563 

644509 

4511563 

467 


First  consider  the  first  two  periods  only,  and  proceed  as 
in  Art.  339.  Having  found  the  first  two  terms  of  the  root, 
consider  them  as  the  tens'  term,  and  proceed  as  before. 


343.      RULE  FOR   FINDING    THE   CUBE  ROOT  OF  A  NUMBER. 

1.  Point  off  the  number  into  periods  of  three  jjJaces 
■each,  beginning  at  the  decitnnl  point  and  counting  to  the 
left  for  integers  and  to  the  right  for  decimals. 

2.  Find  the  largest  cube  in  the  left-hand  period  and 
jtlace  its  root  at  the  right.  Subtract  the  cube  from  the 
left-hand  period  and  annex  the  second  period  to  the 
■vemainder. 


EVOLUTION.  333 

3.  Find  three  times  the  sqtiare  of  the  first  term  of  the 
root,  annex  two  meros,  and  place  it  at  the  left  as  a  trial 
difisor.  Compare  it  tvith  the  second  dividend  and  place 
the  Quotient  as  the  second  term  of  the  root. 

4.  Find  three  times  the  product  of  the  first  and  second 
terms  of  the  root,  anncjc  one  zero,  and  write  it  under  the 
trial  dirisor.  Sqtiare  the  second  term  of  the  root  and  write 
the  result  tinder  tlie  preceding  prodiict»  Find  the  sum  of 
these  three  results  and  jniiltiply  it  by  the  second  term  of 
the  root.  Subtract  the  product  thus  found  from  the  partial 
dividend  and  to  the  remainder  annex  the  next  period. 

5.  Find  three  times  the  square  of  the  root  already  found, 
annex  tujo  zeros,  and  tvrite  it  at  the  left  as  a  trial  divisor. 
Find  the  third  term  of  the  root  and  complete  the  divisor 
as  before. 

Notes.  (1)  If  the  number  is  not  a  cube,  the  work  may  be  continued 
to  any  extent  by  reducing  the  last  remainder  to  thousandths,  millionths, 
etc.,  by  annexing  periods  of  three  zeros  each. 

(2)  If  the  denominator  of  a  common  fraction  is  not  a  cube,  both  terms 
may  be  multiplied  by  some  number  that  will  make  the  denominator  a  cube. 
Why  is  it  more  important  to  have  the  denominator  a  cube  than  the 
numerator  ? 

(3)  If  the  partial  dividend  at  any  time  will  not  contain  the  trial  divisor, 
write  a  zero  in  the  root,  annex  two  zeros  to  the  preceding  divisor  as  a 
new  divisor,  and  proceed  as  before.  Show  the  reason  for  annexing  these 
zeros  by  going  through  the  work  with  zero  as  a  term  of  the  root. 

(4)  The  rules  for  squaring  numbers  from  25  to  100  will  be  found  con- 
venient in  forming  the  trial  divisor  for  the  third  term  of  the  root. 

(5)  Mixed  numbers  should  usually  be  changed  to  improper  fractions  or 
to  mixed  decimals. 

344.    The  following  rules  will  be  found  convenient : 

RULES  FOR  SQUARING   NUMBERS  FROM   25  TO  100. 
1.    To  square  mimbers  frotn  25  to  50, 

(a)  Subtract  25  from  the  number. 

(b)  Subtract  the  remainder  from  25. 

(c)  Square  the  last  result,  and 

(d)  Add  it  to  the  first  result  considered  as  hundreds. 


334  NEW  ADVANCED  ARITHMETIC. 

Illustrative  Example.     42-  =  ? 

(a)  17.       (h)  8.       (c)  64.       (d)  1764. 

2.  To  SQ-uarc   numbers  from  50  to   75. 

(a)  Subtract  50  from  the  number, 

(b)  Add  the  result  to  25. 

(c)  Square  the  first  result,   and 

(d)  Add  it  to  the  second  result  considered  as  hundreds. 

Illustrative  Example.     69^  =  ? 

(a)  19.       (&)  44.       (c)  361.       (c?)  4761.  • 

3.  To  square  numbers  from  73  to  lOO. 

(a)  Subtract  the  number  from  lOO. 

(b)  Subtract  the  first  result  from  the  number. 

(c)  Square  the  first  result,  and 

(d)  Add  it  to  the  second  result  considered  as  hundreds. 

Illustrative  Example.     88"  =  ? 

(a)  12.       (6)  76.       (c)   144.       {d)   7744. 

Note.  Inquisitive  students  -n-ill  desire  to  find  the  reasons  for  these 
rules. 

The  first  is  an  application  of  this  formula:  (a  —  b)-  —  a-  —  Inh  -\-  ^A 
a  here  represents  50,  and  b  the  difference  between  50  and  the  given 
number. 

The  second  is  an  application  oi  [a  +  h]"  =  a-  +  2ab  -\-  b-;  the  letters 
being  used  as  in  the  first. 

The  third  employs  {a  —  b)-;  a  representing  100,  and  b  the  difference 
between  100  and  the  given  number. 

345.      PROBLEMS. 

1.  -^86004573  =  ?     What  is  the  remainder? 

2.  v^.0685  =  ?     Carry  root  to  thousandths. 

3.  ^2  =  ?     Carry  root  to  hundredths. 

4_    /y^:5  =  ?  6.    15.625^  =  ? 

5.    '^^3698.400375"=  ?  7.    \/Ui  =  ? 


EVOLUTION.  335 

8.  ^2Uji  =  'i  16.  -^1  =  ? 

9.  '^^U  =  ?  17.  ^/^  =  ? 

10.  ^27189441343  =  ?  18.  \^^7  =  ? 

11.  34328125^  =  ?  19.  '^QA  =  ? 


12.  194104539»  =  ?  20.  V^0l25  =  ? 

13.  223648543^  =  ?  21.  '^^  =  ? 

14.  736314327'  =  ?  22.  9^  =  ? 


15.  V1003003001  =  ?        23.  Vf  =  ? 

Note.     Teachers  should  dictate  many  problems  uutil  pupils  can  work 
rapidly  and  accurately. 

24.  A  cubical  block  contains  96  feet  of  lumber.     How 
many  inches  long  is  each  edge  ? 

25.  A  cubical  cistern  holding  3,600  gallons  is  how  deep? 

26.  Find  the  dimensions  of  a  cubical  bin  whose  capacity 
is  2,000  bushels. 

346.     1.   The  method  of  extracting  the  cube  root  of  a  number  may  be 
illustrated  bv  tlie  use  of  blocks. 


2.  V50653  =  ? 

A  cubical  figure  contains  50,6.53  cubic  inches.  What  is  the  length  of 
one  edge  ? 

3.  Tiie  number  of  inches  in  the  edge  is  a  two-place  number.  The 
largest  cube  in  .50,000  is  27,000.  27,000  inch-cubes  will  foim  a  figure 
each  of  whose  edges  is  30  inches.  This  27,000  is  "  the  cube  of  the  tens," 
the  first  part  of  the  expansion  shown  in  Art.  338,  2.  23,653  inch-cubes 
remain  with  which  to  enlarge  the  figure.  The  size  must  be  increased  in 
such  a  way  as  to  retain  the  form  of  a  cube.  Since  the  length,  breadth, 
and  thickness  are  equ;\l,  the  additions  must  be  made  to  three  adjacent 
faces,  and  must  l)e  equal.  If  a  layer  of  cubes  were  placed  upon  one  face, 
30  X  30  =  900  cubes  would  be  required.  Since  three  such  additions  are 
to  be  made,  2,700  cubes  would  be  needed  to  make  the  additions  one  inch 
thick  on  each  face.  We  thus  find  the  illustration  of  "  three  times  the 
square  of  the  tens  as  a  trial  divisor."  It  is  called  a  "  trial  divisor  "  be- 
cause we  wish  to  ascertain  how  many  such  additions  may  be  made  to  each 
face  with  the  remaining  blocks,  and  yet  leave  enough  to  fill  out  the  figura 


336 


NEW  ADVANCED  ARITHMETIC. 


The  iudications  are  that  7  such  layers  may  be  added  to  each  face.  This 
would  use  up  3  X  30-  X  7  olocks.  The  expression  is  "  three  times  the 
square  of  the  teus  by  the  uuits,"  or  the  second  part  of  the  expausion 
siiown  iu  Art    338,  2. 

3  X  30-  X  7  n=  18,900.  23,653  —  18,900  =  4,753.  the  number  of  inch- 
cubes  remaining. 

The  figure  is  now  37  inches  wide,  37  inches  long,  and  37  inches  high, 
but  is  not  a  cube,  since  additions  are  still  to  be  made  along  three  edges 
and  on  one  corner.  Each  of  these  additions  to  the  edges  must  be  7  inches 
by  7  inches  and  30  inches  high.  These  will  require  3  X  30  X  T''^  =  4,410 
blocks.  This  expression  is  "  three  times  the  teus  by  the  square  of  tiie 
units,"  or  the  third  part  of  the  expausion  showu  iu  Art.  333,  2. 

4,753  —  4,410  =  343,  the  remaining  blocks. 

An  unfilled  corner,  7  inches  by  7  inches  by  7  inches,  remains.  Since 
343  blocks  are  needed  to  make  the  figure  a  cube,  it  is  clear  that  50,653  is 
the  cube  of  37.  The  343  is  "  the  cube  of  the  uuits,"  or  the  last  part  of  the 
expansion  shown  in  Art  338,  2. 


Fig.  1.     A  Cube. 


FiG.  2.     Additions  for  Cube. 


Ililllilllilil 


Fig.  3.    Additions  made  to  Cube.  Via.  5.    Small  Cube  for  Comer. 


EVOLUTION. 


337 


EXPLANATION  OF  THE  FIGURES. 

Fig.  1  represents  the  cube  formed  from  27,000  blocks. 
Fig.  2  represents  the  three  additions  made  to  the  faces  of  Fig.  1. 
Fig.  3  represents  the  figure  resulting  from  the  three  additions  to  the 
faces  of  Fig.  1 . 

Fig.  4  represents  the  three  additions  made  to  the  edges,  and 
Fig.  5  the  final  addition  made  to  the  corner  to  complete  the  cube. 

Apply   this    method    of    illustration    to    Problems    1-6, 
Art.  339. 


347.      MISCELLANEOUS   PROBLEMS. 

1.  "What  is  the  area  of  a  piece  of  ground  arranged  in  the 
form  of  a  triangle,  the  length  of  one  side  being  124:  rods, 
and  the  distance  to  the  vertex  of  the  opposite  angle  being 
86  rods? 

2.  How  many  feet  of  lumber  are  there  in  a  two-inch 
plank  in  the  form  of  an  isosceles  triangle,  18  feet  long  and 
14  inches  wide  at  the  base? 

3.  Find  the  area  in  acres  of  the  irregular  field  A  B  C  D  E. 


Distance : 
'AD  =  4.0  rd. 
EG=4:6  rd. 
AB  =4:U  rd 
DF=S7  rd. 
BII=SS^  rd 
DC  =  52'rd. 


4.  A  board  in  the  form  of  a  rhombus  is  19  inches  on 
each  side  and  10  inches  high.  Draw  the  figure.  "What  is 
its  area?  Compare  it  with  a  square  that  is  19  inches  on 
each  side. 

Note.    A  Ehombus  is  an  oblique-angled  equilateral  parallelogram. 


338  NEW  ADVANCED  ARITHMETIC. 

5.  A  field  in  the  form  of  a  trapezoid  contains  23^  acres. 
One  of  its  parallel  sides  is  95  rods  and  the  other  65  rods 
long.     What  is  its  width? 

XoTE  1.    A  Trapezoid  is  a  quadrilateral  only  two  of  whose  sides  are 
parallel.     Its  area  is  equal  to  the  prod- 
_B  uct  of  its  height  aud  half  the  sum  of 
its  parallel  sides. 

Note  2.  Drawing  G  E  parallel  to 
B  C  makes  G  B  C  E  a  parallelogram. 
AGE  =  EE1);  hence  A  B  C  D  is 
equivalent  to  GBCE,  and 

EC  =  \(^AB+  CD). 

6.  What  is  the  area  of  a  circular  pond  whose  diameter  is 
15  rods? 

7.  What  is  the  circumference  of  a  circus  ring  whose  area 
is  872f  square  yards? 

8.  What  is  the  area  of  a  field  in  the  form  of  a  trapezoid, 
the  parallel  sides  being  42  rods  and  88  rods  respectively, 
and  the  distance  between  these  sides  being  36  rd.  3  yd.  ? 

9.  A  cubical  box  contains  79,507  cubic  inches.  What  is 
the  length  of  each  edge? 

10.  A  cubical  tank  has  a  capacity  of  7 60 J  gallons.  Find 
the  length  of  one  side.     (Approximate.) 

11.  A  cubical  cistern  is  9  feet  deep.  What  will  it  cost  to 
construct  it  at  75  cents  a  barrel  (31^  gallons)  ? 

12.  A  corn-bin  whose  width  equals  its  height  is  4  times 
as  long  as  it  is  high.  If  it  will  hold  3,200  bushels  of  shelled 
corn,  what  is  its  length? 

13.  A  building  is  36  feet  wide.  If  the  attic  is  9  feet 
high,  what  is  the  length  of  the  rafters,  allowing  for  a  projec- 
tion of  18  inches? 

14.  If  the  area  of  an  equilateral  triangle  is  12  square  feet, 
what  is  the  area  of  a  similar  triangle  each  of  whose  sides  is 
twice  as  long?     Four  tunes  as  long?     7^  times  as  long? 


EVOLUTION.  339 

Note.  Similar  plane  figures  are  those  having  the  same  shupe  ;  e.  g.,  a 
square  is  similar  to  a  square,  a  circle  to  a  circle. 

Two  similar  surfaces  are  to  each  other  as  the  squares  of  like  lines. 

15.  A  circle  whose  diameter  is  4  feet  is  what  part  of  a 
circle  whose  radius  is  6  feet?     8  feet?     9  feet?     \Q\  feet? 

16.  A  circle  whose  diameter  is  6  feet  is  how  many  times 
a  circle  whose  diameter  is  6  inches?  9  inches?  li  feet? 
4  feet? 

17.  A  cube  2  feet  high  is  how  many  times  one  that  is  1 
foot  high  ? 

Note.  Similar  solids  are  those  having  the  same  shape ;  e.  g.,  a  cube  is 
similar  to  a  cube,  a  sphere  to  a  sphere. 

Similar  solids  are  to  each  other  as  the  cubes  of  like  lines. 

18.  A  sphere  whose  diameter  is  1  inch  is  what  part  of  a 
sphere  whose  diameter  is  2  inches?  4  inches?  7  inches? 
1  foot? 

19.  How  many  2-inch  spheres  contain  as  much  volume  as 
a  4-inch  sphere  ?    an  8-inch  sphere  ?    a  foot  sphere  ? 

20.  Compare  the  volumes  of  earth  and  moon,  the  diameter 
of  the  former  being  about  8,000  miles,  and  of  the  latter 
about  4,000  miles.     Compare  their  surfaces. 

21  o  Compare  the  volumes  of  sun  and  earth,  the  diameter 
of  the  former  being  about  880,000  miles.  Compare  their 
surfaces. 

22.  If  a  cannon-ball  weighs  36  pounds,  what  will  one 
weigh  whose  diameter  is  3  times  as  great? 

23.  Height  of  cylinder  6  feet ;  diameter  of  base  21  feet. 
Find  convex  surface.     Find  entire  surface.     Find  volume. 

24.  The  volume  of  a  cylinder  is  72  cubic  feet.  The 
diameter  is  4  feet.  What  is  the  convex  surface?  Entu'e 
surface? 

25.  The  volume  of  a  cylinder  is  196.35  cubic  feet.  Its 
height  is  10  feet.  What  is  the  area  of  the  base  of  a  similar 
cylinder  whose  volume  is  27  times  as  great? 


340  NEW  ADVANCED  ARITHMETIC. 

Note.  Tlie  surface  of  a  sphere  is  3.1416  tiroes  the  square  of  its 
diameter. 

26.  What  is  the  surface  of  a  sphere  whose  diameter  is 
4  inches?     2^  feet?     1  yard? 

27.  What  is  the  area  of  the  earth's  surface,  counting  the 
diameter  7,912  miles? 

28.  The  surface  of  a  sphere  that  is  1  inch  in  diameter  is 
what  part  of  the  surface  of  a  sphere  whose  diameter  is  2 
inches?     4  inches?     6 J  inches? 

29.  What  is  the  diameter  of  a  sphere  whose  surface  is 
31,416  square  miles?  201.0624  square  feet?  337  square 
inches  ? 

30.  Compare  the  surfaces  of  two  spheres  whose  diameters 
are  as  3  to  7.     As  2  to  5.     As  2^  to  3^. 

31.  The  volume  of  a  sphere  is  3.1416  times  J  of  the  cube 
of  the  diameter.  What  is  the  volume  of  a  sphere  whose 
diameter  is  6  inches? 

32.  What  is  the  approximate  interior  diameter  of  a  sphere 
that  will  hold  a  gallon? 

33.  What  is  the  weight  of  a  sphere  of  gold  2  inches  in 
diameter  ? 

34.  What  is  the  weight  of  a  sphere  of  stone  whose  diame- 
ter is  30  inches,  the  stone  weighing  three  times  as  much  as 
water? 

35.  Steel  weighs  about  7.84  times  as  much  as  water. 
What  is  the  weight  of  a  hollow  steel  sphere  whose  interior 
diameter  is  12  inches,  the  shell  being  ^  of  an  inch  thick? 

36.  The  State  in  which  you  live  is  what  part  of  the  sur- 
face of  the  earth  ? 

37.  What  is  the  diameter  of  a  circle  whose  area  is  equal 
to  the  area  of  tlie  State  in  which  you  live  ? 

38.  What  is  the  surface  of  a  sphere  whose  diameter  is 
one  foot? 


EVOLUTION. 


341 


39.  If  the  above  sphere  represents  the  world,  for  how 
many  square  miles  does  a  square  inch  of  its  sui'f ace  stand  ? 
What,  then,  is  the  scale? 

40.  What  is  the  diameter  of  a  circle  that  will  represent 
the  area  of  the  State  in  which  you  live,  on  the  above  globe? 

41.  What  is  the  weight  of  a  cast-iron  street-roller,  8  feet 
long  and  5  feet  in  diameter,  the  shell  being  2  inches  thick, 
cast-iron  being  7.2  times  as  heavy  as  water? 


348.     THE    CONE. 

1.  A  Cone  is  a  solid  formed  by  the  revolution  of  a  right 
triangle  upon  its  base  or  perpendicular  as  an 
axis. 

2.  A  cone  is  one  third  of  a  cylinder  having 
the  same  base  and  altitude  (height). 

3.  The  convex  surface  of  a  cone  is  equal 
to  one  half  the  product  of  the  circumference 
of  the  base  and  the  slant  height.      Why  ? 

PROBLEMS. 

1.  Height,  20  inches;  radius,  15  inches.  Find  convex 
surface  and  volume. 

2.  Slant  height,  10  inches;  height,  8  inches.  Find  volume 
and  entire  surface. 

3.  Cii'cumference  of  base,  75.3984  feet;  slant  height,  20 
feet.     Find  volume  and  entire  surface. 

4.  The  radius  of  a  cone  is  7  inches  and  its  altitude  15 
inches.  What  is  the  convex  surface  of  a  similar  cone  whose 
radius  is  10  inches?     What  is  its  volume? 

5.  What  is  the  weight  of  a  steel  paper-weight  in  the 
form  of  a  cone,  the  diameter  of  the  base  being  3  inches  and 
the  height  4  inches,  steel  being  7.83  times  as  heavy  as 
water  ? 

23A 


342 


A'iiTr  ADVANCED  APdTHMETIC. 


3i9.  1.  The  Frustum  of  a  cone  is  that  portion  of  the 
cone  included  between  the  base  and  a  plane 
passing  through  the  cone  parallel  to  the 
base. 

2.  The  convex  surface  of  a  frustum  of  a 
cone  is  a  Trapezoid.  How  may  its  area  be 
found  ? 

3.  The  frustum  of  a  cone  is  equivalent  to  three  cones 
having  the  altitude  of  the  frustum,  and  whose  bases  are  the 
upper  base  of  the  frustum,  the  lower  base,  and  a  m.ean  pro- 
portional between  them. 

4.  To  find  the  volume  of  a  frustmn  of  a  cone,  find  the 
sum  of  til e  upi>er  base,  the  lotrer  base,  and  the  square  root 
of  their  product,  and  mnJtiply  the  result  by  one  third  of 
the  altitude  {distance  betiveen  the  bases). 

PROBLEM. 
Find  convex  surface  of  a  frustum  of  a  cone,  the  radius  of 
whose  upper  base  is  10  inches,  of  the  lower  base  15  inches, 
and  whose  altitude  is  21  inches.     Find  the  volume. 


350.     PRISMS    AND    PYRAMIDS. 

1.  A  Right  Prism  is  a 
solid  two  of  whose  faces 
are  equal  and  parallel 
polygons,  and  the  rest  of 
whose  faces  are  rectan- 
gles. 

2.  The  volume  of  a 
right  prism  is  found  in  tlie 
snmo  way  as  the  volume 
of  a  right  parallelopiped. 

3.  A  pyramid  is  a  solid 
lliat  is  bounded  by  a  poly- 
gon called  the  base,  and  three  or  more  triangles  that  meet  at 
a  point. 


EVOLUTION.  343 

4.  The  volume  and  convex  surface  of  a  pyramid  are  found 
in  the  same  way  as  the  volume  and  convex  surface  of  the 
cone. 

5.  Define  a  Frustum  of  a  Pyramid.     Art.  349,  1. 

Tell  how  to  find  the  volume  and  convex  surface  of  a  frus- 
tum of  a  pyramid.     Art.  349,  2,  3. 

Describe  the  above  figures.     Find  a  frustum  of  a  pyramid. 

PROBLEMS. 

1.  What  is  the  volume  of  a  prism  whose  base  is  11  square 
feet  and  height  10  feet?  Of  a  pyramid  of  the  same  dimen- 
sions? 

2.  What  is  the  volume  of  a  prism  whose  altitude  is  6  feet 
and  whose  base  is  a  triangle  each  of  whose  sides  is  2  feet? 
of  a  pyramid  of  the  same  dimensions  ? 

3.  What  are  the  volume  and  convex  surface  of  a  pyramid 
whose  altitude  is  b\  feet,  and  whose  base  is  a  square  the 
diagonal  of  which  is  8  feet? 

4.  Find  the  convex  surface  and  volume  of  a  frustum  of  a 
square  pj-ramid,  a  side  of  the  lower  base  being  3^  feet,  of  the 
upper  base  2^  feet,  and  whose  height  is  4^  feet. 

5.  Find  the  volume  of  a  pyramid  whose  base  is  an  equi- 
lateral triangle,  each  side  measuring  6  feet,  and  whose  alti- 
tude is  24  feet. 

6.  Compare  the  volume  of  this  pyramid  with  that  of  a 
pyramid  of  the  same  height,  but  whose  base  is  6  feet  square. 


344  NEW  ADVANCED  ARITHMETIC. 


351.     GENERAL    REVIEWS. 
I. 

1.  Define  Multiplicatiou,  Denominator,  Decimal  Fraction, 
Interest,  Sphere. 

2.  What  common  fractions  can  be  changed  to  pure  deci- 
mals?    Why? 

3.  8243  ^  ?     ^^80964. 73807  =  ? 

1.18      3.64 

4.  Smiplify  ;j5-2  X  ^^- 

5.  Derive  a  rule  for  dividing  by  a  fraction. 

6.  Find  the  interest  on  $824.60,  at  7%,  from  June  12, 
1887,  to  Sept.  5,  18'JO. 

7.  How  many  gallons  of  water  will  a  cylindrical  cistern 
hold  whose  diameter  is  '1\  feet  and  depth  9i  feet? 

8.  Change  §  of  a  square  mile  to  integers  of  lower  denom- 
inations. 

9.  What  is  the  cost  of  the  lumber  and  posts  to  fence  the 
N.  W.  \  of  the  S.  \  of  the  N.  E.  ^  of  a  section  with  a  four- 
board  fence,  posts  8  feet  apart  and  costing  23  cents  each, 
fencing  at  $18.50  per  M.? 

10.  What  is  the  cost  of  a  40-day  draft  for  $580,  exchange 
being  \%  premium,  and  interest  at  6%  ? 

11.  What  is  the  difference  between  seven  hundred  and 
two  thousandths  and  seven  hundred  two  thousandths  ? 

II. 

1.  Divide   3,744    into   three    parts  in  the    proportion   of 

h  h  h 

2.  Define  Ratio,  Proportion,  Commission,  2sotation, 
Division. 


GENERAL  REVIEWS.  345 

3.  How  many  square  feet  of  sheet-iron  \  of  an  inch  thick 
can  be  made  from  a  cylindrical  shaft  20  feet  lono-  and  4 
inches  in  diameter? 

4.  If  52  men  can  dig  a  canal  355  feet  long,  GO  feet  wide, 
and  8  feet  deep  in  15  days,  how  long  will  a  canal  be  that  is 
45  feet  wide  and  10  feet  deep  which  45  men  can  dig  in  25 
days  ? 

5.  What  is  the  selling  price  of  an  article  bought  at  a  dis- 
count of  33^%,  and  sold  at  an  advance  of  12i%,  yielding  a 
gain  of  $8? 

6.  How  many  gallons  of  liquid  will  a  hollow  sphere  hold 
whose  inner  diameter  is  22  inches? 

7.  A  45-foot  ladder  placed  between  two  poles  reaches  one 
of  them  24  feet  from  the  ground,  and  the  other  28  feet. 
How  far  apart  are  they? 

8.  What  is  the  difference  of  time  between  two  places 
whose  difference  of  longitude  is  46°  18'  46"? 

9.  If  -/j  of  an  acre  of  land  cost  S33|,  what  will  36|  acres 
cost? 

10.  A,  B,  C,  and  D  together  own  a  tract  of  land  2  miles 
square.  A  owns  |-  as  much  as  B ;  B  §  as  much  as  C ;  C  | 
as  much  as  D.     How  many  acres  has  each? 

III. 

1.  Define  Antecedent,  Consequent,  Mean  Proportional, 
Complex  Fraction,  Compound  Interest. 

2.  What  is  the  average  time  for  the  payment  of  $600  due 
in  3  months,  $480  due  in  5  months,  $390  due  in  8  months, 
and  $850  due  in  14  months? 

3.  Find  the  1.  c.  m.  of  174,  485,  14,065. 

4.  Give  the  rule  for  "  pointing"  the  quotient  in  division 
of  decimals,  and  give  the  explanation  of  it. 

5.  Change  18  mi.  124  rd.  4  yd.  to  feet,  and  use  three 
forms  of  analysis  in  the  reductions. 


346  NEW  ADVANCED  ARITHMETIC. 

6.  A  public  square  is  surrounded  by  a  walk  2  rods  wide. 
The  area  of  the  walk  is  an  acre.  What  is  the  area  of  the 
square?     Make  a  figure. 

7.  For  what  amount  shall  a  90-day  note  be  made  that  the 
proceeds  shall  be  $358.60,  interest  at  7%  ?   (Bank  Discount.) 

8.  State  the  three  general  problems  of  percentage. 

9.  -^j;  is  what  per  cent  of  ^5?     Give  an  analysis. 

10.  A,  B,  and  C  form  a  partnership,  and  make  a  gross 
gain  of  816,440.  A  invests  85,000  for  12  months ;  B,  89,000 
for  16  months;  C,  87,100  for  6  months.  The  total  expenses 
were  84,110,  which  they  agreed  to  share  equally.  What  was 
each  partner's  share  of  the  net  gain? 

IV. 

1.  Define  Subtraction,  Minuend,  Subtrahend,  Remainder, 
Partition. 

2.  Divide  83,600  by  37^,  and  give  the  analysis.  Multiply 
564  by  83 J,  and  give  the  analysis. 

3.  Give  tests  of  divisibility  by  3,  4,  8,  9,  11. 

4.  If  a  school-room  is  15  feet  high,  how  many  square  feet 
of  floor  must  it  have  to  furnish  60  persons  300  cubic  feet  of 
air?  If  the  length  is  to  the  breadth  as  4  to  3,  what  will 
each  be? 

5.  A  cubic  bin  with  a  square  bottom  holds  164,025  cubic 
inches.  Depth  is  to  width  as  9  to  5.  What  is  the  depth? 
The  width  ? 

6.  40  men  agree  to  do  a  piece  of  work  in  50  days,  but 
after  working  9  hours  a  day  for  30  days  only  half  the  work 
is  completed.  How  many  additional  men  must  be  employed 
to  finish  the  work  on  time  by  putting  in  10  hours  a  day? 

7.  What  is  the  present  worth  of  a  non-interest-bearing 
debt  of  8728.40,  due  in  3  years,  7  mouths,  and  19  days, 
money  being  worth  7  %  ? 

8.  V860473. 02986  =  ? 


GENERAL   REVIEWS.  347 

9.  Find    the   value    of    the    foUowhig   lumber   at   §21    a 
thousand : 

4  6x8  sills,       16  feet  long. 

26  2  X  8  joists,     18    "       " 

30  2  X  4  studs,     22    "       " 

18  2  X  6  rafters,  20    "       " 

10.  Find    the    premiums    on    the    following    policies    of 
insurance  : 

62,100,  at  l\%.         82,800,  at  2\% .         63,150,  at  |%. 


1.  "When  do  you  conclude  that  a  number  is  prime?    Why? 

2.  Give  the  demonstration  of  the  test  of  divisibility  by  9. 

3.  The  proceeds  of  a  note  for  8265.50,  discounted  on 
June  12,  1891,  at  7  %,  were  8263.     When  was  the  note  due? 

4.  The  proceeds  of  a  90-day  note  for  8480  were  8470.08. 
What  was  the  rate  of  discount?     (With  grace.) 

5.  Bought  goods  for  8729.  What  must  they  be  marked 
that  the  merchant  may  fall  10%,  lose  10%  on  bad  debts,  and 
still  gain  10%  ? 

6.  A  circular  piece  of  laud  16  feet  in  diameter  is  to  be 
divided  into  3  equal  parts,  the  inner  part  being  a  circle,  and 
the  second  and  third  parts  being  circular  strips.  What  is 
the  diameter  of  the  inner  circle  ?  What  is  the  width  of  each 
of  the  circular  strips? 

Note.  What  is  the  area  of  the  whole  circle  ?  What  is  the  diameter 
of  the  inner  circle  ? 

7.  Give  two  ways  of  changing  a  common  fraction  to  a 
decimal.     Change  1%-^  to  a  decimal,  and  explain  each  stop. 

8.  A  can  do  a  piece  of  work  in  2|  days ;  B,  in  3*  days ; 
and  C,  in  4i  days.  How  long  would  it  take  them  to  com- 
plete the  job  working  together?  If  ZQ>  is  paid  for  the  whole 
work,  what  is  the  share  of  each? 


348  ^'EW  ADVANCED  ARITHMETIC. 

9.  A  note  of  $1,200,  dated  April  1,  1886,  and  bearing  in- 
terest at  8% ,  bad  the  following  indorsements  :  Sept.  12,  1886, 
$130.  March  20,  1887,  $240.  Aug.  24,  1888,  $325.  What 
was  due  April  1,  1890? 

10.  When  it  is  noon  in  Boston,  what  time  is  it  at  San 
Francisco  ? 

11.  At  what  rate  must  4%  bonds  be  purchased  to  yield 
o^7o  on  the  investment? 

12.  Write  the  tables  of  long,  square,  and  cubic  measures ; 
also  of  Troy,  Apothecaries',  and  Avoirdupois  weights. 

VI. 

1.  A  man  travelled  at  the  rate  of  3  mi.  165  rd.  4  yd.  2  ft. 
an  hour.     How  far  did  he  go  in  36  hours? 

2.  How  many  revolutions  will  a  wheel,  whose  diameter  is 
4i  feet,  make  in  rolling  3  miles? 

3.  A  hollow  brass  sphere,  whose  diameter  is  4  inches, 
weighs  I  as  much  as  a  solid  sphere  of  the  same  size  and 
material.     How  thick  is  the  shell? 

4.  Express  the  ratio  of  a  pound  Troy  to  a  pound  Avoirdu- 
pois ;  of  an  ounce;  of  a  cubic  foot  to  a  bushel;  of  a  quart 
liquid  measure  to  a  quart  dry  measure. 

5.  Solve  the  following  by  compound  proportion  :  If  15 
men  in  12  days  of  10  hours  each  can  dig  a  ditch  180  rods 
long,  6  feet  wide,  and  4  feet  deep,  how  many  hours  a  day 
must  10  men  work  to  dig  a  ditch  200  rods  long,  8  feet  wide, 
and  2  feet  deep  in  10  days? 

6.  Add  :     5  A.  120  sq.  rd.  21  sq.  yd.  6  sq.  ft. 

12   "     96    " 
22  "     83    " 

17  "     74    " 

7.  Find  the  interest  on  $1,580,  at  7i%,  from  Dec.  18, 
1889,  to  May  1,  1891. 


18 

(,( 

u 

7 

25 

u 

(( 

4 

28 

(( 

a 

8 

GENERAL   REVIEWS.  349 

8.  What  is  the  cost  of  a  UO-day  draft  ou  London  for  £850, 
exchange  being  $4.86,  and  interest  5%  ?     (With  grace.) 

9.  What  is  the  value  of  a  pile  of  wood  360  feet  long,  12 
feet  wide,  and  6  feet  high,  at  §3.20  a  cord? 

VII. 

1.  Put  the  following  items  into  the  form  of  a  receipted 
bill: 

R.  D.  Smith  bought  of  Cole  Bros.,  Newark,  N.  J.,  on 
June  1,  1892,  16  yards  silk,  @  $1.85.  June  12,  56  yards 
cotton  cloth,  @  9  cents.  June  15,  8  yards  broadcloth,  @ 
S2.25.  June  20,  24  yards  carpet,  @  96  cents.  July  1,  31 
yards  matting,  @  40  cents.     July  10,  5  sets  curtains,  @  $3.85. 

2.  Find  the  g.  c.  d.  of  1127  and  6581. 

3.  A  man  bought  a  house  and  lot  for  $5,088.  3  of  the 
cost  of  the  house  was  |  of  the  cost  of  the  lot.  What  was 
the  cost  of  each? 

4.  At  $4.75  a  cord,  what  is  the  cost  of  the  following  piles 
of  cord  wood : 

a.  18  feet  long,  6    feet  high. 

b.  23    "       "      b\    "       " 

c.  17    "       "      7      "      " 

5.  Change  2  rd.  4  yd.  2  ft.  to  the  decimal  of  a  mile. 

6.  What  must  be  the  rate  of  taxation  in  a  town  to  yield 
a  net  return  of  $16,660,  if  the  real  estate  is  assessed  at 
$531,000,  the  personal  property  at  $200,182.80,  7  %  of  the 
tax  being  uncollectible,  and  the  collector's  commission  being 

2%? 

7.  In  what  time  will  $469.50  yield  $36.80,  at  7%  ? 

8.  What  is  the  volume  of  a  sphere  whose  diameter  is  7^ 
inches  ? 

9.  If  a  block  of  stone  18  inches  long,  4  inches  wide,  and 
2  inches  thick,  weighs  12  lb.  15  oz.,  what  is  the  weight  of  a 


350  NEW  ADVANCED  ARITHMETIC. 

block  of  the  same  material  2J  feet  long,  2  feet  wide,  and  9 
iuclies  thick? 

10.  What  is  the  volume  of  the  frustum  of  a  cone  the 
diameters  of  whose  bases  are  respectively  28  inches  and  16 
inches,  and  whose  height  is  30  inches? 

VIII. 

1.  Anal^'ze  each  of  the  following: 

a.  2  is  what  part  of  5^?  b.  f  is  what  part  of  \\? 
c.    .015^  is  what  part  of  .63? 

2.  Add/y,    2of  51,    §^|,   f  X31. 

3.  Give  the  rrle  for  "pointing  '  the  product  in  Multipli- 
cation of  Decimals,  and  explain  it. 

4.  Change  61,368  seconds  to  integers  of  higher  denomina- 
tions, and  give  the  analysis  for  two  reductions. 

5.  f  of  an  inch  is  what  part  of  a  rod?     Analyze. 

2i 

6.  "What  number  multiplied  by  jy  '"'i^l  gi'^'e  ~  foi'  ^  product? 

7.  Change  y\  of  a  mile  to  integers  of  lower  denominations. 

8.  "What  will  it  cost  at  24  cents  a  square  yard  to  plaster  a 
hall  AQ^  feet  wide,  82  feet  long,  and  24:  feet  high,  no  allow- 
ance being  made  for  openings? 

9.  Bought  42  shares  of  stock  at  lOoi^.  Received  a  4% 
dividend,  and  sold  the  stock  at  103  J-.  The  gain  was  what  per 
cent  of  the  investment? 

10.  "What  principal  will  amount  to  Si, 690  from  Jan.  12, 
1890,  to  June  17,  1892,  at  7%  interest? 

11.  A  railroad  train  moves  a  mile  in  65  seconds.  "What 
is  its  rate  per  hour? 

IX. 

1.  What  is  the  area  of  an  equilateral  triangle  each  of 
whose  sides  is  42  rods? 


GENERAL   REVIEWS.  351 

2.  Find  the  number  of  acres  in  the  following  tracts  of 
land  : 

a.  N.  i  of  S.  W.  I  of  a  section. 

b.  S.  iof  X.  W.  i-  of  S.  E.  1. 

c.  S.  W.  \  of  S.  E.  \  of  N.  AV.  ^.  Make  a  figure  showing 
each  tract. 

3.  A  is  26  rd.  4  yd.  north  of  C.  B  is  13  rd.  S^  yd.  east 
of  C.     AVhat  is  the  distance  from  A  to  B? 

193  3i  I3i 

4.  From    "•*  take  the  sum  of  §  X  —  and  |  X  — ?,  and 

divide  the  result  by  2|f . 

5.  How  many  -iO-gallon  barrels  will  a  cylindrical  cistern 
hold,  the  diameter  of  which  is  9i  feet,  and  whose  depth  is 
10  feet? 

6.  Which  is  the  better  investment,  5%  bonds  at  98,  or 
4%  bonds  at  95?  Give  the  rate  per  cent  of  interest  on  each 
investment. 

7.  What  is  the  per  cent  of  gain  if  ^  of  an  article  is  sold 
for  ^^  of  its  cost? 

8.  Explain  the  rule  for  "  pointing  "  a  number  to  ascertain 
the  number  of  terms  in  its  cube  root. 

9.  Bought  15  railroad  bonds  at  H%  discount,  brokerage 
1|%.  For  what  must  a  90-day  note  be  drawn,  interest  at 
8  7c ,  to  obtain  the  amount  of  the  purchase  at  a  bank  ?  (With 
grace.) 

10.  Find  the  cube  root  of  3  to  within  .001. 

X. 

1.  Define  a  ratio.  Define  each  term.  What  is  the  differ- 
ence between  a  ratio  and  a  fraction?  Define  a  proportion, 
and  each  term.  By  what  principle  is  any  term  of  a  propor- 
tion found  when  three  are  given?  » 

2.  A  can  do  a  piece  of  work  in  6  days;  B,  in  7  ;  and  C, 
in  8.     In  what  time  can  they  do  it  working  together? 


352  NEW  ADVANCED  ARITHMETIC. 

3.  What  is  the  capacity  of  a  hollow  sphere  whose  outside 
diameter  is  15  inches,  and  whose  walls  are  ^  of  an  inch 
thick? 

4.  A  man  bought  four  articles  for  $896.45.  For  the  first 
he  gave  821  more  than  for  the  second;  for  the  second, 
$62.50  more  than  for  the  third;  for  the  third,  $81.75  more 
than  for  the  fourth?     What  did  each  cost? 

5.  Sold  160  acres  of  land  at  $87.50,  commission  2|-%. 
Directed  the  agent  to  iuv^est  the  proceeds  in  5%  bonds  at  98, 
reserving  his  commission  at  2%,  and  returning  the  surplus  of 
less  than  $100.  How  many  bonds  did  he  purchase,  and  how 
much  did  he  return  ? 

6.  V|=.?      \/i  =  ? 

7.  The  interest  on  two  sums  of  money  for  4  j^ears  and  8 
months  at  6%  was  $256.  §  of  the  first  sum  equalled  the 
second.     What  was  each? 

8.  A  steamer  can  sail  10  miles  an  hour  with  the  current, 
and  5  miles  an  hour  against  it.  AVhat  is  the  rapidity  of  the 
current?  How  long  a  trip  up  stream  and  down  can  it  make 
in  6  hours? 

9.  Find  the  volume  of  a  cone  whose  altitude  is  15  inches, 
and  radius  of  base  3h  inches. 

10.  A  commission  merchant  sold  goods  at  2%  brokerage. 
He  invested  the  proceeds  at  2%,  reserving  his  commission. 
His  commissions  amounted  to  $149.  What  was  the  amount 
of  the  first  sale? 

XI. 

1.  Give  the  laws  of  the  Roman  Notation. 

2.  Explain  the  philosophy  of  "  pointing  "  for  the  extrac- 
tion of  the  cube  root. 

3.  How  many  rolls  of  paper  are  needed  to  cover  the 
walls  and  ceiling  of  a  room  16  feet  by  18  feet,  and  11  feet 
high,  deductions  being  made  for  three  windows  3  ft.  2  in,  by 


GENERAL   REVIEWS.  353 

7  ft.  4  in.,  2  doors  3  ft.  by  8  ft.  2  in.,  and  a  10-inch  base 
board  ? 

4.  Find  the  compound  interest  on  8483.93  for  4  yr.  5  mo. 
13  d.,  at  6%,  compounded  semi-annually. 

5.  A  room  is  15  feet  by  18  feet,  and  10  feet  high.  What 
is  the  length  of  a  line  extending  from  an  upper  corner  diago- 
nally through  the  room  to  an  opposite  lower  corner? 

6.  What  is  the  diameter  of  a  circle  containing  20  acres  of 
land?  What  is  the  area  of  a  strip  18  feet  wide,  lying  next 
to  the  circumference  and  reaching  around  the  field  on  the 
outside  ? 


8.  On  a  certain  farm  the  barn  cost  f  as  much  as  the 
house,  and  the  house  ^  as  much  as  the  land.  The  tenant 
raised  3,600  bushels  of  corn  and  3,000  bushels  of  oats.  The 
landlord  received  %  of  the  corn  and  \  of  the  oats  for  rent 
Corn  sells  for  44  cents,  and  oats  for  3H  cents.  The  land- 
lord's income  was  ^\l\%  of  his  investment.  Find  cost  of 
barn,  house,  and  farm. 

9.  How  many  bushels  of  corn  in  the  ear  will  a  crib  hold 
that  is  46  feet  long,  8  feet  wide,  and  10  feet  high,  counting 
the  bushel  at  |  of  true  capacity  ? 

XII. 

1.  What  is  the  face  of  a  60-day  note  the  proceeds  of  which 
are  $2,654.38  when  discounted  at  a  bank  at  7%  ?  (With  grace.) 

2.  A  and  B  can  do  a  piece  of  work  in  24  days.  A  can  do 
I  as  much  as  B.     In  how  many  days  can  each  do  it  alone? 

3.  A  rectangular  field  contains  12^  A.  Its  width  is  f  of 
its  length ;  what  is  the  distance  around  it  ? 

4.  A,  B,  and  C  were  partners  in  business.  A's  capital 
was  §  of  B's,  and  B's  was  f  of  C's.  A's  capital  was  in  8 
months;  B's,  9  months;  C's,  10  months.  Their  net  gains 
were  §2,674;  what  was  the  share  of  each? 


354  NEW  ADVANCED  ARITHMETIC. 

5.  An  agent  sold  a  house  and  lot  for  his  principal.  After 
reserving  his  commission  of  2%  for  selling  and  2%  for  buy- 
ing, he  invested  the  remainder  in  corn  at  51  cents  a  bushel. 
His  total  commissions  were  $400 ;  how  many  bushels  of  corn 
did  he  buy? 

6.  How  many  gallons  of  water  will  a  hollow  sphere  hold 
whose  interior  diameter  is  3^  ft.  ? 

7.  A  piece  of  land  in  the  form  of  a  trapezoid  is  120  rd. 
between  its  parallel  sides,  one  of  which  is  45  rd.  long,  and 
the  other  60  rd.  long.  What  is  the  land  worth  at  §62.50  an 
acre? 

8.  A  merchant  sold  a  customer  7  pieces  of  cloth,  each 
containing  50  yd.  He  made  a  reduction  of  20%  from  the 
retail  price,  and  a  further  reduction  of  5%  for  cash.  The 
retail  price  was  40%  above  cost.  He  received  $532.  What 
was  the  retail  price  per  yard? 

9.  A  certain  note  for  $600  is  dated  June  1,  1890.  It  is 
due  two  years  from  date,  and  bears  6%  interest.  What 
should  be  paid  for  it  Sept.  1,  1890,  in  order  that  the  invest- 
ment shall  yield  10%  per  annum? 

xni. 

1.  Define  Division.  Define  a  Common  Fraction.  Effect 
of  multiplying  numerator  and  denominator  by  the  same 
number?     Explain  fully. 

2.  Give  and  fully  explain  the  rule  for  the  multiplication  of 
Decimal  Fractions. 

3.  What  sum  of  money  put  at  interest  for  3  years  7 
months  18  days,  at  6%,  will  amount  to  $648.36? 

4.  A  tank  can  be  filled  by  keeping  one  pipe  open  4  hours, 
or  by  keeping  a  second  pipe  open  for  5  hours.  The  tank 
has  a  pipe  by  means  of  which  it  can  be  emptied  in  2^  hours. 
In  what  time  will  the  tank  be  filled  if  the  three  pipes  be  left 
open? 


GENERAL   REVIEWS.  355 

5.  Change  ^j  of  a  mile  to  integers  of  lower  denominations. 

6.  A  farmer  has  a  -iO-aere  field  in  the  form  of  a  square. 
He  has  it  planted  in  corn,  the  rows  being  3  feet  G  inches 
apart.  The  first  row  is  3  feet  from  the  line.  How  far  does 
he  walk  in  plowing  it  once,  taking  a  row  at  a  time? 

7.  A,  B,  and  C  form  a  partnership  for  one  year.  A  put 
in  83,000  for  the  first  six  months,  when  he  withdrew  $1,000. 
B  put  in  83,000  at  first,  and  when  A  withdrew  he  made  the 
deficit  in  their  joint  capital  good.  C  put  in  85,000  for  the 
year.  Their  net  gain  was  82,750.  What  was  each  one's 
share  ? 

8.  I  spent  25%  of  my  money,  33  J  %  of  the  remainder, 
and  8^-%  of  the  remainder.  I  then  had  8550.  How  much 
did  I  have  at  first? 

9.  A  man  was  paying  rent  at  the  rate  of  815  a  mouth. 
He  borrowed  an  amount  from  a  Building  Association  which 
enabled  him  to  build  a  house  as  good.  He  made  a  monthly 
payment  of  $18  for  six  years,  when  his  house  was  paid  for. 
How  much  more  than  his  rent  did  the  house  cost  him,  count- 
ing interest  on  his  money  at  6%  ? 

10.  A  house  is  30'  by  40'.  The  cistern  connected  with 
its  roof  is  cylindrical  in  shape,  9  feet  deep,  and  has  an 
average  diameter  of  10  feet.  At  the  end  of  a  rain  the 
cistern  was  found  to  be  half  full.  How  many  inches  of  rain 
had  fallen? 

11.  A  man  left  $10,000,  to  be  invested  for  his  three  sons, 
aged  12,  15,  and  18,  at  4%  compound  interest.  He  directed 
that  the  money  should  be  so  divided  that  the  children  would 
receive  equal  amounts  when  21  years  of  age.  How  much 
was  set  aside  for  each  one? 

12.  AVhat  is  the  rate  of  interest  received  on  an  investment 
in  A\  bonds  purchased  at  106? 

13.  At  what  rate  must  3^%  bonds  be  bought  to  yield  4% 
on  the  investment? 


2. 

9  +  x=  16 

3. 

a^-3  =  12 

4. 

12  — a;  =    2 

356  NEW  ADVANCED  ARITHMETIC. 


352.     ALGEBRAIC    QUESTIONS. 

Find  what  number  x  stands  for,  if 

1.    X  +  7  =  10       5.    Ix  =  21  9.  3x  +  ox  =  32 

6.  3.«  +  5  =  29      10.  9.C  —  2.?;  =  35 

7.  7  +  2x  =  17      11.  7x  +  3x  —  5x  =  30 

8.  17  —  ■2x  =  13      12.    5x—2x-\-  11-17 

13.  If  the  base  of  this  rectangle 
is  b  units,  the  altitude  a  units,  ex- 
press the  sum  of  the  base  and 
altitude. 

14.  Express  the  difference  of 
base  and  altitude. 

15.  Express  the  length  of  the 
perimeter  (entire  boundary)  o 

16.  Express  the  area. 

17.  Express  what  part  the  altitude  is  of  the  base. 

18.  Express  what  part  the  base  is  of  the  perimeter. 

353.     THE  LITERAL  NOTATION. 

1.  Numbers  are  often  represented  by  letters,  as  in  the 
questions  above.  Any  letter  may  stand  for  any  number, 
but  in  any  particular  problem  a  letter  must  stand  for  the 
same  number  throughout. 

2.  A  letter,  or  a  combination  of  letters,  standing  for  a 
number  is  called  an  algebraic  expression. 

3.  The  numbers  expressed  by  algebraic  expressions  may 
be  sums,  differences,  products,  quotients,  powers,  or  roots. 

A  power  is  the  product  of  equal  factors ;  the  degree  of  the  power  is 
determined  by  the  number  of  equal  factors,  and  is  indicated  by  a  small 
figure  or  letter  (written  above  and  to  the  right),  called  the  exponent  or 
index. 


ALGEBRAIC  QUESTIONS.  357 

Thus,  23  =2X2X2  =  8.     Eight  is  the  third  power  of  two. 

A  root  of  a  iinniher  is  oue  of  the  equal  factors  whose  product  is  the 
given  number.  2  is  the  3d  root  of  8 ;  5  is  the  2d  root  of  25.  Roots  are 
indicated  by  the  radical  sign,  V.  Tlie  index  of  the  root  is  a  numeral 
written  above  tlie  radical  sign. 

-^16  =  2.     The  fourth  root  of  16  is  2. 

Y  r/^  —  d.     The  third  root  of  d  third  power  is  d 

2d  powers  and  3d  powers  are  called  squares  and  cubes. 

2d  roots  and  3d  roots  are  called  square  roots  and  cube  rootSr 

The  index  of  square  roots  is  usually  not  written. 

\/9  =  3.     The  square  root  of  9  is  3. 

4.  In  reading  algebraic  expressions  involving  more  than 
one  letter,  it  is  often  best  to  name  the  kind  of  number  be- 
fore reading  it ;  thus, 

(1)  a  +  3  6,  the  sum  of  a  and  3  J,  or  a  plus  3  h. 

(2)  6  a  —  3  c,  the  difference  of  6  a  and  3  c,  or  6  a  minus  3  c. 

Note.  Such  expressions  for  sums  and  differences  are  called  binomials. 
The  expressions  connected  by  tlie  signs  +  or  —  are  called  terms.  7  a  and 
5  c-  are  the  terms  of  the  binomial  7  a  —  5  c.  An  expression  of  three  terms 
is  a  trinomial,  as  3  a  +  2  c  —  5  .r.  Any  expression  of  two  or  more  terras 
may  be  called  a  polynomial,  although  the  name  is  usually  applied  to  ex- 
pressions of  more  than  three  terms.  A  monomial  contains  one  term  only. 
Polynomials  are  sometimes  called  compound  expressions.  Compare  com- 
pound numbers  with  polynomials. 

(3)  a  X  (6  —  ■),  or  a  (b  —  c),  the  product  of  a  and  the 
binomial  b  minus  c,  or  a  times  the  binomial  b  minus  c. 

(4)  (3x  —  a)  -f-  c,  the  quotient  of  the  binomial  3  x  minus 
a  divided  by  c. 

■VT  .  3  .r  —  a 

Note.     A  quotient  written  is  called  a  fraction. 

(5)  a  —  (.r  —  ?/)3,  a  minus  the  cube  of  the  binomial  x  mi- 
nus y. 

(6)  a  ■—  {x  —  y^),  a  minus  the  binomial  x  minus  y  cubed.. 

(7)  Va^  —2  c,  the  cube  root  of  a  squared  minus  2  c 

24A 


358  NEW  ADVAXCED  ARITHMETIC. 

(8)  ('V^a)'^  —  2  c,  the  square  of  the  cube  root  of  a  minus  2  c 

(9)  ^a^  —2  c,  the  cube  root  of  the  binomial  a  squared 
minus  2  c. 

,^^.     V  {3(t  —  b)  (2x  +  y),  the  fraction,  the  square  root 
a  +  X  —  y  of  the  product  of  the  binomials 

3  a  minus  b  and  2  a;  plus  y  divided  by  the  trinomial  a  plus  x 
minus  y. 

354.     EXERCISES  IN  LITERAL  NOTATION. 
Write  as  algebraic  expressions : 

1.  The  sum  of  a  squared  and  the  square  root  of  c. 

2.  Five  times  the  binomial  the  fourth  root  of  x  cubed 
minus  7  a  b. 

3.  Five  times  the  fourth  root  of  x  cubed  minus  lab. 

4.  Five  times  the  fourth  root  of  the  binomial  x  cubed 
minus  lab. 

5.  a  plus  the  fraction  b  divided  by  c  plus  d. 

6.  a  plus  the  fraction  b  divided  by  the  binomial  c  plus  d. 

7.  The  fraction  a  plus  b  divided  by  the  binomial  c  plus  d. 

8.  The  fraction  a  plus  b  divided  by  c  plus  d. 

9.  The  square  of  the  fraction  a  minus  c  divided  by  x. 

10.   The  product  of  3  times  the  sum  of  x  and  y  by  the 
sum  of  3  x  and  y. 

355.     EVALUATION  OF  ALGEBRAIC  EXPRESSIONS. 

To  evaluate  an  algebraic  expression  is  to  find  its  numer- 
ical value ;  thus : 

If  a  =  1,   &  =  2,   c  =  3,   re  =  10,    what  is   the   value   of 
/babe  —  2a  +  hx\^ 
V  3x  — 2c 


ALGEBRAIC   QUESTIONS,  359 

FORM. 

.^s  ^5abc  —  2a+h  x\^ 

^^  \        3x~^^^        )   ^ 

.  /5-1-2-3-2-1  +  2-  lOy  __ 

^"^  \  3  ■  10-2  ■  3  )   ~ 

(;))  (2)^  = 

(G)  8 

Description.  Substituting  in  expression  (1)  the  numer 
ical  values  of  the  letters  we  have  exp.  (2).  Performinc 
indicated  multiplicatious  we  have  exp.  (3)*.  Performino-  ad- 
ditions and  subtractions  we  have  exp.  (4).  Dividing  we  have 
exp.  (5).     Cubing  the  quotient,  as  indicated,  we  obtain  8. 

356.      PROBLEMS   IN    EVALUATION 

a  =  1,  &  =  2,  c  =  3,  cZ  =  10,  .T  =  0. 

1.  3a&  +  ca;+4d  6.    lOc— (6d  — 4ac) 

2.  ^/hc-V  d  7.    10c  — &d  — 4ac 

3.  (2fZ-&c)2  8.    ^4&  +  2rZ-a 

4.  (3  a  +  &)  .r  +  2  a  c  d  9.    ^^^4  6  +  2  d  —  a 

5.  a.T+ &.'r -f  cd  10.    (d  —  by 

a-{-2b  +  d'  11.    (d^  —  b^ 

357.      EQUATIONS. 

Problems.  1.  A  boy  has  four  times  as  many  white  mar- 
bles as  brown  ones.  Of  both  he  has  30.  How  many  of 
each? 


360  NEW  ADVANCED  ARITHMETIC. 

Let  X  =  number  of  brown  marbles. 

Then,  since,  etc.         4  x  =  number  of  -white  marbles. 
And  z  +  4  X  =  number  of  both. 

But  30  =  number  of  both. 

Therefore  x  +  4  z  =  30. 

ox  =  30. 

X  =    6,  no.  of  brown  marbles. 
4x  =  24,  no.  of  white  marbles. 


358.       DEFINITIONS. 

(1)  An  equation  is  a  statement  in  mathematical  symbols 
that  two  expressions  stand  for  the  same  number. 

.r  +  4 X  =  30  in  the  problem  above  is  an  equation,     x  +  Ax 
and  30  each  stand  for  the  total  number  of  marbles. 

(2)  The  numbers  of  an  equation  are  the  two  equivalent 
expressions,  x  +  Ax  and  30,  in  the  equation  above. 

(3)  Equations   are  used  to  find  the  value  of   unknown 
numbers  represented  by  a-,  y,  z,  etc. 

(4)  To  solve  an  equation  is  to  find  the  value  of  the  un- 
known number  involved. 

2.    In  a  school  of  45  pupils  there  are  7  more  girls  than 
boys.     How  man}"  of  each? 

Let  X  =  number  of  boys. 

Then,  since,  etc.,  x  +  7  =  number  of  girls. 

And  X  +  X  +  7  =  number  of  pupils  in  school. 

But  4.5  =  number  of  pupils  in  school. 

Therefore  x  +  x  +  7  =  4.5. 

2  X  4-  7  =  45. 
2  X  =  38. 
X  =  19,  number  of  boys. 
X  4-  7  =  26,  number  of  girls. 

Show  that  in  the  above  solution  we  have   proceeded  in 
accordance  with  the  following  truths : 


ALGEBRAIC   QUESTIOyS.  361 


359.     Axioms. 

(1)  Things  equal  to  the  same  thing  are  equal  to  each 
other. 

(2)  If  equals  be  added  to  equals,  the  sums  are  equal. 

(3)  If  equals  be  subtracted  from  equals,  the  remainders 
are  equal. 

(1)  If  equals  be  multiplied  by  equals,  the  products  are 
equal. 

(5)  If  equals  be  divided  by  equals,  the  quotients  are 
equal. 

3.  Six  times  John's  age  exceeds  four  times  his  age  by 
22  years.     How  old  is  he? 

^'oTE.  Let  X  =  the  number  of  years  in  John's  age.  If  we  say  let  x  = 
John's  age,  we  treat  x  as  a  mere  quantity  of  time,  not  as  a  number  of 
time-units. 

4.  821,000  is  divided  among  three  children  so  that  the 
first  receives  twice  as  much  as  the  second,  the  second  twice 
as  much  as  the  third.     What  is  the  share  of  each? 

XoTE.     Let  X  =  no.  of  dollars  in  the  share  of  the  third. 

5.  Thomas.  Richard,  and  Heury  have  72  marbles.  Thomas 
has  twice  as  many  as  Richard.  Henry  has  twice  as  many  as 
both  the  others.     How  many  has  each? 

6.  How  old  am  I,  if  three  times  my  age  four  years  ago 
exceeds  twice  my  present  age  by  27  years? 

7.  Equal  weights  of  sugar  and  flour  were  bought  for  63 
cents.  The  sugar  cost  5  cents  per  pound,  the  flour  2  cents. 
How  many  pounds  of  each? 

8.  The  perimeter  of  a  rectangular  field  80  rods  long  is 
280  rods.     What  is  its  width  ? 

9.  The  perimeter  of  a  rectangular  field,  twice  as  long  as 
wide,  is  180  rods.     What  is  its  length? 


362 


NEW  ADVANCED  ARITHMETIC. 


V 


iO.  It  takes  70  feet  of  border  to  enclose  a  square  room. 
W^hat  are  its  dimensious  ? 

11.  A  room  27  feet  wide  and  x  feet  long  requires  99 
square  yards  of  matting.     What  is  the  value  of  x? 

12.  A  Sunday-school  collection  in  dimes,  nickels,  and 
cents  amounted  to  200  cents.  There  were  three  times  as 
many  nickels  as  dimes  and  five  times  as  many  cents  as  nickels. 
How  many  of  each? 

13.  Grace  is  5  years  older  than  May.  May  is  two  years 
older  than  Ethel.  The  sum  of  their  ages  is  42  years.  What 
is  the  age  of  each  ? 

f^      14.    A  father  is  four  times  as  old  as  his  son.     Five  j^ears 
ago,  he  was  seven  times  as  old.     What  is  the  father's  age? 

Let  X  —  no.  of  years  in  son's  age. 

Then  (why  ?)  4  x  =  no.  of  years  in  father's  age. 

"          "  X  —  o  =  no.  of  years  in  son's  age  5  years  ago. 

"          "  7  (z  —  .5)  =  no.  of  years  in  father's  age  5  years  ago. 

"          "  4:  X  —  5  =  no.  of  years  in  father's  age  5  years  ago. 

Hence  (1)  7  (x  -  5)  =  4  x  -  5. 

(2)  7x-.3o  =  4x-5. 

(3)  7  X  =  4  x  -  5  +  35. 

(4)  7x-4x  =  35-5. 

(5)  3x^30. 

(6)  X  =  10. 

(7)  4  X  =  40,  no.  of  years  in  father's  age. 
What  was  added  to  each  member  of  (2)  ?    What  was  subtracted 

from  each  member  of  (3)? 

Note.  If  a  term  contain  ,i  "  numerical  "  factor,  and  one  or  more  literal 
factors,  the  numerical  factor  is  called  the  coefficient  of  the  term.  Terms 
containing  the  same  literal  factors  are  called  like  terms.  In  the  terms  7  t, 
3  a'^  b,  7  and  3  are  the  coefficients.  3  a  x^  and  5  a  x-  are  like  terms.  3  d-  b 
and  1  ah-  are  unlike  terms.     Why  ? 

I;i  describing  the  solution  of  the  equation  in  Problem  14  we  may  say: 
Performing  indicated  multiplication  in  Eq.  (1 )  we  have  Eq.  (2).     Add- 
ing 35  toeach  member  of  Eq.  (2)  we  obtain  Eq.  (3).     Subtracting  -ix  from 
each  member  of  Eq.  (3)  we  obtain  Eq.  (4).     Collecting  like  term.s  in  (4)j 
we  obtain  Eq.  (5)     Dividing  each  member  of  Eq.  (5)  by  the  coefficient  of 


ALGEBRAIC  QUESTIONS.  363 

X,  3,  we  obtain  Eq.  6.  Multiply iug  each  member  of  (6)  by  4  we  get  Eq. 
(7).  (Adding  35  to,  and  subtracting  4r  from,  each  member  of  Eq.  2  we 
obtain  Eq.  5.) 

15.  A  man  of  35  is  7  times  as  old  as  his  son.  In  how 
many  years  will  he  be  twice  as  old? 

Let  X  —  no.  of  years  hence  when  the  father's  age 

will  equal  twice  the  son's  age. 
Then  5  +  x  =  no.  of  years  in  son's  age  at  required  time ; 

and  2  (5  +  a;)  =  no  of  years  in  father's  age  at  required  time. 

But  35  +  a;  =  no.  of  years  in  father's  age  at  required  time. 

Hence  (1)  2  (5  +  x)  =  35  +  x. 

(2)  10  +  2  a;  =  35  +  x. 

(3)  2x-x  =  35-10. 

Note.  In  subtracting  10  and  x  from  each  member  of  Eq.  (2),  we  cause 
each  of  these  terms  to  pass  to  the  other  member  of  the  equation  with 
change  of  sign.  This  transfer  of  a  term  to  the  opposite  member  with 
change  of  sign  is  called  transposition. 

16.  A  has  8  dollars  more  thafi  B.  After  paying  B  12 
dollars,  A  has  only  ^  as  many  as  Bo  How  much  had  each 
at  first? 

17 o  John  has  40  marbles  more  than  Fred.  After  giving 
Fred  50,  John  has  only  i  as  many  as  Fred. 

18.  A  debt  of  $102  is  paid  with  an  equal  number  of  ten- 
dollar,  five-dollar,  and  two-dollar  bills.  How  many  bills  were 
paid  in  all? 

19.  In  paying  27  cents  for  an  article,  I  tendered  some 
dimes  and  received  an  equal  number  of  cents  as  change. 
How  many  dimes  did  I  tender? 

20o  Harry  and  Walter,  62  miles  apart,  ride  towards  each 
other.  Harry,  starting  at  9  a.m.,  rides  2  miles  per  hour 
faster  than  Walter,  who  started  at  8  A.  m.  They  meet  at 
nooDo     What  is  the  rate  of  each? 

21.  Take  some  number,  double  it,  add  20,  divide  by  2, 
take  away  the  first  number;  you  have  10  left.  Why  is  this? 
If  you  had  added  30,  instead  of  20,  how  many  would  you 
have  left? 


364  NEW  ADVANCED  ARITHMETIC. 

22.  Take  some  number,  multiply  by  6,  add  30,  divide  by 
3,  subtract  4,  divide  by  2,  take  away  the  first  number;  you 
have  3  left.     Explain. 

360.    EQUATIONS  CONTAINING  FRACTIONS. 

1,  I  of  what  number  =  4  ? 

2.  I  of  what  number  =3? 
3o  f  of  what  number  —  6? 
4o  j\  of  what  number  ■--.  10? 

5o  ij  of  a  number  +  J  of  the  same  number  =  21.  What 
is  the  number? 

6.  f  of  a  number  —  ^  of  the  same  number  =10.  What 
is  the  number? 

7o   f  of  a  number  =  7.     What  is  twice  the  number? 

8o    I  of  a  number  =11.     What  is  four  times  the  number? 

X 

9.  -  =  5o     Fmd  value  of  x. 
o 

By  what  number  must  we  multiply  each  member  to  obtain 
the  value  of  x  ? 

10.  -—  =  36. 

5     . 

By  what  number  must  we  multijjly  each  member  to  obtain 
the  value  of  4a;?      Will   10  do?     What  other  multipliers? 
What  axiom  is  involved? 
x      4x 

"•    3  +  X  =  ''■ 

Analysis.  To  make  the  first  fraction  integral  we  must  m.ultiplj  it  by 
3,  or  some  multiple  of  3 ;  to  make  tlie  second  fraction  integral  we  must 
multiply  it  by  5,  or  some  multiple  of  5;  to  preserve  the  equality  of  the 
members  we  must  multiply  both  by  the  same  multiplier.  15  is  tlie  least 
common  multi])le  of  3  and  .5.  Multiplying  the  first  fraction  by  15  (first 
by  3  to  suppress  the  denominator,  and  then  by  5)  we  have  5x.     Multiply- 


ALGEBRAIC   QUESTIONS.  365 

ing  the  second  fraction  by  15  (first  by  5,  and  then  by  3)  we  have  I2r. 
Multiplying  the  second  member  by  15,  we  have  255.  Our  equation  now 
stands : 

5x  +  I2x=  255. 

Note.    This  process  of  transforming  a  fractional  equation  into  an  inte- 
gral equation  is  called  clearing  of  fractions. 

Solve  : 

,„     4.x      ^x       ^^  3       4       5 

12.    —  +  -—  =  31.  23.   -  +  -  +  -  =  1. 


5         4 


13.   ^+10  =  ^-1.  2*-  i^-  +  i^-^^-  =  25„ 

^                    ^  25.  %x-\-  %x~}x^  82. 

3a;       2x  _  3  ^  g 

'''     4         3   -'•  26.  ^/__^  +  ^.34. 

4^^ +5  .^^  +  8.  28.  2fa;  =  105. 


16. 


'  '  29.    3lx  =  48. 

z    .  z       z 


17,    1  +  ^  +  -  =  39.  30.    Ux-  2|  .r  =  45. 

2       3       4 


31.    51  a;  —  32  a;  =  44„ 

4^2~5~'"'  32.    5i  (.T  — 3)  — ^  =  0. 


18.    -  +  -__  =  22.  ,,     -,   ,         o,       2a; 


32;       33  z 


3  a; 


^^-    T  -  T0~  ""  ^^*  33.  X-  -^  2  a;  +  —  =  36. 

20     ^  ^      2/  ,   '^^       99  4  a;       7  -  a-       , 

'°-    T-2  +  5  =  2^'  34.  --+-^  =  53. 

3a; +  4       2a;-8_  10. ^       .r  _  3        ,^ 

21-    ■ — ^ +  — o =  ^-       35.  -^  +  -—     ^    >0 

2a;      3a;+  4  3a;— 4 


366 


NEW  ADVANCED  ARITHMETIC. 


361.      DEFINITIONS. 

1.  If  the  members  of  an  equation  are  alike  in  form,  or  if 
they  are  reducible  to  the  same  form,  the  equation  is  called 
an  identical  equation,  or  an  identity.  Thus,  9  =  9,  5  -f  2 
—  4  =  3,  and  5  a;— 7  =  3  x'— 7  +  2  x  are  identities. 

2.  A  solution  is  verified  by  substituting  for  x  in  the 
given  equation  the  value  of  x,  as  found  in  the  solution, 
and  performing  all  indicated  operations  in  each  member. 
If  the  equation  reduces  to  an  identity,  -  the  solution  is 
correct. 

Illustration  : 

3  +  2a;      7a;+6_18a; 
5       "^  ~Tl       ~^^ 

66  +  44  ic  +  70  a;  +    60  =  198  a; 

114a;  +  126  =  198a; 

126  =    84a; 

I  =  X. 

Verification  : 

f  +  6      18  .  f 


3  +  2  .  I      7_ 


11 


3+3 


10^  +  6 

n" 


+     :.^—  = 


6      16| 

5^   11 


10 

27 
10 

27 
10 


6      3  _27 
5^2  ""10 

27_27 
10~10 

3.  If  upon  substituting  for  x  its  supposed  value  the  equa- 
tion becomes  an  identity,  the  value  of  x  is  said  to  satisfy  the 
e(iuation. 


ALGEBRAIC  QUESTIONS.  367 

362.      PROBLEMS. 
Solve  and  verify : 

1.  Divide  90  into  two  such  parts  that  one  shall  be  3^ 
times  the  other. 

2.  Divide  100  into  two  such  parts  that  one  shall  be  2^ 
times  the  other. 

3.  A  horse  was  sold  for  $80,  at  a  gain  of  I  of  the  cost. 
What  was  the  cost? 

4.  A  is  12  years  older  than  B.  \  of  A's  age  =  ^  of  B's. 
WhaJt  is  the  age  of  each  ? 

5.  If  to  John's  age  there  be  added  its  half,  its  third,  and 
its  fourth,  the  sum  is  25  years.     What  is  his  age? 

6.  If  to  Mary's  age  there  be  added  its  half,  its  third,  and 
its  fifth,  the  sum  is  2^  times  her  age.     What  is  her  age? 

Query.     What  is  the  matter  with  the  foregoing  problem  ? 

7.  If  to  A's  age  there  be  added  its  double,  its  half,  and 
its  third,  the  sum  lacks  7  years  of  four  times  his  age.  What 
is  his  age  ? 

8.  A,  B,  and  C  received  $162  for  digging  a  ditch.  A 
dug  4  rods  to  B's  3  rods  and  C's  2  rods.  What  pay  should 
each  receive? 

9.  Two  barrels  contain  respectively  42  and  50  gallons  of 
oil.  After  drawing  the  same  amount  from  each,  the  first 
contained  §  as  much  as  the  second.  How  much  was  drawn 
from  each? 

10.  A  campaign  pole  84  feet  high  broke  at  such  a  point 
that  the  top  was  f  of  the  stump.  What  was  the  height  of 
the  stump? 

11.  A  campaign  pole  100  feet  broke  at  such  a  point  that 
the  top  was  6  feet  longer  than  the  stump.  What  was  the 
length  of  the  stump  ? 


368  NEW  ADVANCED  ARITHMETIC. 

12.    The  sum  of  two  numbers  is  s,  the  difference  d.    What 
are  the  numbers? 

Let  X  —  the  smaller  number. 

Then  x  +  d  —  the  greater  number, 
and  -X  +  X  -}-  d  =  s 
'i  X  -}-  d  ^  s 

2x  =  s  —  d 

smaller  number. 


2 

s  —  d       2d       s  -\-  d  , 

X  +  d  =  — 1 — —  =  — - — ?  greater  number. 

^  Jt  £d 

Note.  In  solving  the  preceding  problem  we  have  solved  every  prob- 
lem in  which  the  sum  and  difference  of  two  numbers  are  given  to  find  the 
numbers ;  for  s  and  d  are  any  numbers. 

By  substituting  the  values  of  s  and  d  in  any  particular 
problem  of  this  type,  we  avoid  a  formal  solution.     Thus : 

13.  The  sum  of  two  numbers  is  42,  their  difference  12. 
What  are  the  numbers? 

s  +  d       42  +  12       54 
Greater  number  =  — 5 —  =  — 9 ~  ^  ~  "^ 

s  —  d 
Smaller  number  =  — - — 


14. 


Sight  problems  : 

Sum. 

Difference. 

(1)   30 

20 

(2)   30 

e 

(3)    22 

8 

(4)   55 

5 

(5)    44 

6 

(6)    23 

7 

42-  12 

2        - 

30 
2 

=  15. 

Sum. 

(  7)    88 
(  8)    33 

Difference. 

12 
3 

(  9)    52 

12 

(10)  26 

(11)  13 

(12)  51 

6 

9 

19 

15.    A  boat  runs  6  miles  per  hour  up-stream,  and  16  mileo 
per  hour  down-stream.     What  is  the  rate  of  the  current? 


ALGEBRAIC   QUESTIONS.  -^69 

16.  John  had  six  marbles  more  than  James.  Harry  had 
none.  After  receiving  \  of  James's  marbles,  and  f  of  John's, 
Harry  had  as  many  as  John  or  James.     How  many  had  each? 

363.     POSITIVE    AND    NEGATIVE    QUANTITIES. 

1.  Quantities  are  sometimes  so  related  that  one  tends  to 
neutralize  or  destroy  the  other.  Thus,  a  rise  in  temperature 
counteracts  an  equal  fall ;  debts  are  opposed  to  assets ; 
traveling  southward  cancels  traveling  northward;  the  force 
of  the  current  destroys  an  equal  force  propelling  the  boat  up 
stream.  To  these  quantities,  opposite  in  kind,  the  names 
positive  and  negative  are  applied.  Either  of  a  pair  of  oppo- 
sites  may  be  called  the  positive  quantity ;  usually  the  more 
familiar  of  the  two  is  so  named. 

2.  The  signs  +  and  — ,  wheu  used  as  signs  of  operation, 
are  read  "plus"  and  "minus;"  when  placed  before  num- 
bers to  show  their  character,  they  are  read  "  positive"  and 
"negative." 

3.  At  -i  A.  M.,  Monda}',  the  thermometer 
stood  at  3°  above  zero.  In  the  succeeding 
12-hour  periods  it  (1)  rose  12°,  (2)  fell  19°, 
(3)  rose  7°,  (4)  fell  15°,  (5)  rose  22°, 
(6)  fell  9°.  What  was  the  sum,  or  result- 
^-+5       ''' t'-       ing  temperature,  Thursday  morning? 

Explanation.  The  initial  temperature, 
3°  above  zero,  is  positive.  Addhig  to  it 
the  various  positive  numbers  (rises),  we 
obtain  +44°.  The  sum  of  the  negative 
,  -^  numbers  (falls)  is  —  43^  This  sum  cancels 
+  43  of  the  positive  sum,  leaving  +  1°  as 
the  total  sum,  or  final  temperature. 

364.     PROBLEMS. 

1.  Seven  boys  pull  at  a  rope ;  three  pull  northward,  ex- 
erting respectively  forces  of  75,  85,  and  G3  pounds;  four 


+ 

3 

+ 

12 

— 

19 

+ 

7 

— 

15 

+ 

22 

- 

9 

+  44 

- 

43 

370  NEW  ADVANCED  ARITHMETIC.     • 

pull  southward  with  forces  of  52,  57,  59,  and  43  pounds. 
The  rope  is  moved  in  what  direction  and  with  what  force? 

2.  In  the  left  scale-pan  of  a  balance  are  two  8-ounce 
weights,  three  4-ounce,  and  five  2-ounce.  In  the  right  pan 
are  two  16-ounce  weights  and  three  2-ounce.  What  must 
be  added  to  the  left  pan  to  produce  equilibrium? 

3.  A  letter-carrier  has  walked  north  7  blocks,  east  3 
blocks,  south  5  blocks,  east  2  blocks,  north  4  blocks,  west 
11  blocks,  south  8  blocks,  east  2  blocks.  He  is  now  how 
far  east  of  his  starting-point?     How  far  south? 

Note.     Mark  eastings  aud  northings  +,  westings  and  southings  — . 

4.  A  surveyor  has  measured  N.  6.22  ch.,  E.  4.17  ch., 
N.  2.12  ch.,  W.  6.96  ch.,  N.  3.16  ch.,  E.  11.25  ch.,  S. 
12.58  ch.,  W.  8.62  ch.,  N.  2.00  ch.  He  is  now  how  far 
north  and  east  of  his  starting-point? 

5.  Add  6  a,  —  3  a,  —  7  a,  —  5  a,  +  3  o,  +  6  a,  —  3  a. 

6.  Add  2aa',  — 4aa;,  -fSacc,  4-6aa-,  —  7aa',  -l-3ax. 

7.  Simplify  3a6  —  6a&  +  4a6  —  5a6  —  7a6-|-r2a6. 

8.  Sunplify  hxy  —  lxy  —  22xy  —  xy-\-2hxy  +  2Qxy-' 
10  xy. 

9.  Add  3a-h5  6,  9a-i-36  —  2  c,  6a  —  4  c,  12  c—  18  a. 

FORM. 


3a 

+  5b 

9a 

+  36- 

2c 

6a 

— 

4c 

-18  a 

+ 

12  c 

86  + 

6c 

10.  Add     5a  —  3. T,    9x  —  5y+2a,    Qx  —  2y—la, 
5  a  +  9  ?/. 

11.  Add  3a6  —  2c-j-c?,    7a&  —  6d,   5c—  10a6  —  4d, 
10d-3c. 


ALGEBRAIC   QUESTIONS.  371 

365.     Subtraction. 

1.  What  is  the  change  iu  temperature  if  the  thermometer 
reading  changes  from  +  75°  to  +  90°  ?  from  +  33°  to 
+  56°  ?     from  —  7°  to  +  8°  ?     from  —  2°  to  +  11°  ?     from 

—  4°  to  —  22°  ?    from  +  3°  to  -  7°  ?     from  +  10°  to  —  1°  ? 

2.  What  must  be  added  to  +  3  to  make  +  10?  to  —  3 
to  make  +10?     to  +  10  to  make  +3?     to  +  10  to  make 

—  3  ?     to  —  3  to  make  —  10  ?     to  —  7  to  make  —  2  ? 

3.  Since  the  sum  of  the  subtrahend  and  difference  equals 
the  minuend,  we  may  define  subtraction  as  the  process  of 
finding  from  two  given  numbers  called  subtrahend  and  minu- 
end a  third  number  called  difference,  which  added  to  the 
subtrahend  produces  the  minuend. 

4.  The  minuend  contains  the  subtrahend  and  the  differ- 
ence. If  we  can  destroy  the  subtrahend  in  the  minuend, 
only  the  difference  will  remain.  To  destroy,  or  cancel,  a 
number,  we  add  an  equal  number  with  opposite  sign.  Hence, 
we  add  to  the  minuend  the  subtrahend  with  changed  sign 
and  obtain  the  difference. 

5.  From  5a  —  7^  +  4c  take  2a  -2h  —  2c. 

If  we  write  5  a  —  7  ^  +  4  c  —  2  « ,  we  have  taken  2  a  from 
the  minuend;  but  we  were  required  to  take  away,  not  2'/, 
but  2  a  dmiinished  by  2  5  and  2  c.  We  have  therefore  taken 
away  too  much,  and  must  add  2  b  and  2  c  to  obtain  the  true 
difference.  Hence, 
5a_7^  +  4c_(2a-2Z'-2c)==5a-7^  +  4c-2«  +  25  +  2c. 

Here  we  see,  as  in  4,  that  to  find  the  difference,  we  must 
change  the  signs  of  the  subtrahend  and  add  to  the  minuend. 

366.     PROBLEMS. 

1.  From  lax  —  2>hx  —  4:cy  take  ^ax  —  2hx  ~  cy^ 

2.  From  5  ??«.  +  3  n  +  12^  take  Sm  —  2n  —  1  p. 

3.  From  4  r/  —  3  />  —  he  take  6  a  +  5  c  —  Ad. 


372  NEW  ADVANCED  ARITHMETIC. 

4.  a  +  b  +  {4:a  —  3b)=? 

5.  9a'—7b  —  6c—(3a  +  6b  —  dc)=? 

6.  17  a  —  12  «  m  —  4  c2  —  (—  4  a  —  14  a  m'^  +  2  c^  =? 
7  3a  +  4^  — 5c  — [2a  +  3^  — (2«  — 6e)]  =? 

8.  5a  —  (ox  —  4:  >/)  —  [5  a  ?/  —  3  a  —  (2  a?  +  3  y)]  =? 

9.  7  a2  _  (7  ^,2  _  3  ,.2^  _  2  c2  —  [4  Z/2  4-  (2  a^  -  3  c^)]  =? 

367.     Multiplication. 

1.  If  a  street-car  ride  costs  m  cents,  what  is  the  cost  of 
?  ridesi''     7  rides?     22  rides?     a  rides?     x  rides? 

2.  A  dime  is  tendered  in  payment  for  car-fare;  b  cents 
are  returned  as  change.     What  is  the  cost  of  the  ride  ? 

3.  In  paying  4  such  fares,  how  many  dimes  are  tendered? 
How  many  cents  cliange  are  returned?  In  paying  in  such 
fares  ? 

Note.  In  the  expression  ?n  (10  —  b)  =  10  m  —  b  m,  we  see  that  the 
signs  of  the  product  are  the  same  as  in  the  corresponding  terms  of  the 
multiplicand. 

4.  A  conductor  starts  on  a  trip  with  two  dollars,  collects 
12  such  fares,  and  refunds  3  fares.  The  number  of  cents  he 
now  has  is  expressed 

200  -I-  12  (10  -b)  -S(10-  b). 

The  sign  before  the  multiplier  shows  what  is  to  be  done  with 
the  product. 

In  receiving  12  fares,  he  receives  12  x  10  cents  and  pays 
out  12  X  b,  or  12  6,  cents.  In  refunding  3  fares,  he  pays 
out  3x10  cents  and  receives  3  b  cents.     Therefore  he  now 

has 

200  +  12  •  10  —  12  ^»  -  3  •  10  +  3  b. 

Note.  The  last  four  terms  are  products.  The  positive  terms,  +12  10 
and  -j-  3  b,  are  the  products  of  factors  with  like  signs;  the  negative  terms, 
—  12  6  and  —  3  •  10,  are  tlie  products  of  factors  of  unlike  signs.  Would 
the  signs  be  as  they  are  if  other  numbers  than  12,  b,  10,  and  3  had  been 
used'' 


ALGEBRAIC   QUESTIONS 


373 


RULE. 
Two  factors  of  like  sign  give  a   positive  product;  ttco 
factors  of  unlike  sign,  a  negative  product. 

368.      PROBLEMSo 

1.  A  conductor  starting  with  e  cents  collects  a  —  b  fares 
of  m  —  X  cents  each.  How  many  cents  has  he  at  the  en-l  of 
the  trip?     What  do  —  h  and  —  x  signify  in  this  problem r 

2.  A  butcher  sold  m  guaranteed  hams  at  a  —  h  cents 
each.  X  hams  were  returned  as  spoiled.  What  were  the 
net  receipts  ? 

3.  A  rectangle  is  m  +  w  units  long,  a  +  x  units  wide. 
What  is  its  area? 

Explain  these  diagrams : 

a  +  b 


a 

( 

25A 


4.    Construct  diagrams  for  (a  +  h)"-  and  (a  —  6;-. 


374  NEW  ADVANCED  ARITHMETIC. 

5.  "VMiat  is  the  area  of  a  square  a  -{•  b  units  on  each  siae: 

6.  What  is  the  volume  of  a  rectangular  solid  b  feet  long, 
a  feet  wide,  and  h  feet  high?  How  man}-  cubic  feet  in  the 
bottom  layer? 

7.  What  is  the  volume  of  a  cube  a  +  6  feet  on  each  edge? 

8.  What  is  tJie  volume  of  a  rectangular  solid  a  +  b  feet 
long,  m  +  n  feet  wide,  x  +  y  feet  high? 

9.  What  is  the  area  of  a  rectaugle  a  -{•  b  feet  long,  a  —  b 
(eet  wide  ? 

10.  What  is  the  area  of  a  rectangle  (3  a  —  x)  by  (4  a  —  3  x)  ? 

11.  (3  a  -  4  a:  +  7  &)  (9  X  -  3  a  +  2  6)  =  ? 

12.  (4  a  —  3  a  6)  (3  a  —  5  6)  (a  +  b)  =  ? 

13.  (a  —  c;)(a  —  2x)  (a  +  a-)  =  ? 

14.  (la  —  6  ax)  (Sa  +  4iax  —  ox)  =  ? 

15.  da-7  {a  +  b)  +  A{a  —  2b)  =  ? 

16.  5{x^—xy)+3x(4:X  —  Dy)—oy{2x   _)_  3y  =   ? 

17.  3  (9ax  -  2by)  -  ba  {bx  +  4y)  +  2y  (3  ^/  +  lOO  =  ? 

369.     Division. 

1.  3  6  dollars  are  paid  for  b  cords  of  wood,  '^liat  is  the 
price  per  cord  ? 

2.  axy  trees  are  planted  in  a  equal  rows.  How  many 
trees  in  each  row? 

3.  The  area  of  a  rectangle  is  ax  +  ay.  It  is  a  units 
long.     What  is  its  width? 

4.  The  area  of  a  rectangle  is  4  ax  —  8 xy.  It  is  4  a-  units 
long.     What  is  its  width? 

5.  The  area  of  a  rectangle  is  6  .r^  +  5  a-  —  6  squ  ire  feet. 
Its  length  is  3  x  —  2  feet,     AVTiat  is  its  width  ? 

A  rectangle  3  a:  —  2  feet  long  and  1  foot  wide  contains  how 
many  square  feet?  A  rectangle  3  a;  —  2  feet  long  and  i  x  feet 
wide?  How  many  more  square  feet  in  the  given  rectangle? 
How  manv  more  "  rows"  of  square  feet  do  they  make? 


ALGEBRAIC   QUESTIONS. 


375 


Zx-2)  6x2  +  5x-G  (2x  +  3 
6  x^  —  4  X 


yx-  d 
9x-6 

Note.  The  first  term  of  the  quotient  is  found  by  dividing  the  first 
term  of  the  dividend  by  the  first  term  of  the  divisor.  The  work  proceeds 
as  in  ordinary  long  division. 

6.    Fiud  the  T\idth  of  the  folloTviug  rectangles  : 
5^/4-3  7^  +  2 


28  1/  +  50  y  +  12 


A  X  -\-  jf 


\2x"  +  Uxy  +  2f 


370.     LAW   OP   SIGNS   IN   DIVISION. 

jlemembering  that  the  dividend  is  the  product  of  the 
divisor  and  quotient,  and  that  a  positive  product  is  the 
product  of  two  factors  of  like  sign,  write  the  quotients 
required. 


376  NEW  ADVANCED   ARITHMETIC. 

+  12                           —ab                          +  2la'^x 
1.    ~  =  5. =  9.    = = 

—  7a 

-Seab'^c 
+  12 

ab  —  ax 

+  a 
bx  —  xy 


4  3 

4-  12 

-3 

-  12 

4-3 

-  12 

6. =  10. 


3. =  7.    =  11. 


+  b 

-bx 

+  b 

-be 

—  c 

+  12 

ax 

8. =         12. 

-3  +  6a-  —x 

From  the  examinatiou  of  the  first  four  problems  we 
conclude : 

If  dividend  and  divisor  are  of  like  sign,  the  quotient  is 
positive. 

If  dividend  and  divisor  are  of  unlike  sign  the  quotient  is 
negative. 

371.    Perform  divisions  as  indicated : 


1. 

a-  4-  2  a6  4-  b' 

8. 

a^  —  x" 

a  +  b 

a  -\-  x' 

a^  -2ab  +  b^ 
a  —  b 

9, 

a*  —  X* 

2. 

a--x^ 

3. 

a-'-b^ 

10. 

x^  +  x-  12 

a  —  b' 

X  —  3 

4. 

x^  —  y^ 
X  —  y' 

11". 

x^  -  7a:  +  12 
a;  — 3 

5. 

x^  4  y^ 

12 

a^  +  Sa-b  4  ^ab'  +  6» 

X  +  y  ' 

a  -{-  b 

€. 

X*  +  xh/  4  y^ 
x^  +  xy  +  y^' 

13. 

(x^  _  3.r  4  2)  (a-  -  3), 
a-2  _  Sec  +  6 

7. 

a* -a-* 

14. 

(^2  _  a-  -  12)  (x  4-  5) 

ALGEBRAIC   QUESTIONS.  377 

a  +  X  '  '  x'^  —  l/'^ 

^6      (X  -  yf  +  ^Xy  ^^       (g-^  +  y2)2  _  ^2^. 

a;^  +  ^f        '  X?-  ^  xy  ^  y"^ 

17    i.^-yy{^-^  h)  20       "'  -  ^' 


APPENDIX.  379 


APPENDIX. 


372.     GREATEST    COMMON   DIVISOR. 

1-  A  divisor  of  a  number  is  one  of  the  integral  numbers 
w^hich  being  multiplied  together  -vyill  produce  that  number. 

2.  Name  all  of  the  divisors  of  each  of  the  following 
numbers  : 

4,  6,  8,  12,  15,  24,  36,  39,  40,  48,  56,  64,  72,  96. 

3.  What  number  will  divide  4  and  6?  9  and  12?' 
10  and  15?  6,  9,  and  12?  15,  18,  and  21?  14,  21,  2S, 
and  35? 

4.  What  do  you  call  a  number  that  will  divide  each  of 
two  or  more  numbers? 

5.  A  common  divisor  of  two  or  more  numbers  is  a 
number  that  is  a  divisor  of  each  of  them. 

6.  Name  all  of  the  common  divisors  of  6  and  12.  Which 
is  the  greatest?  Of  8,  16,  and  24,  which  is  the  greatest? 
Of  16,  24,  and  32,  which  is  the  greatest?  What  is  such  a 
number  called  ?     Define  it. 

7.  The  greatest  common  divisor  of  two  or  more  numbers 
is  the  greatest  number  that  is  a  divisor  of  each  of  them. 

8.  Examine  these  groups  of.  numbers  and  find  of  what 
the  greatest  common  divisor  is  the  product  in  each  case. 

9.  The  greatest  common  divisor  of  tvro  or  more  numbera 
is  the  product  of  their  common  prime  factors. 

Prove  the  preceding  statement. 


380  NEW  ADVANCED  ARITHMETIC. 

373.    PRINCIPLES. 

1.  Any  number  is  divisible  by  each  of  its  prime  factors 
and  by  the  product  of  any  number  of  them. 

2.  The  product  of  any  of  the  common  prime  factors  of 
two  or  more  numbers  is  a  common  divisor  of  the  numbers. 

3.  The  product  of  all  of  the  common  prime  factors  of 
two  or  more  numbers  is  theii'  greatest  common  divisor. 

Find  the  g.  c.  d.  of  21,  42,  and  63. 

FORM. 

21  =  3  X  7 

42  =  2  X  3  X  7 
63  =  3  X  3  X  7 

Explanation.  3  is  a  prime  factor  of  each  of  these  num- 
bers. 7  is  also  a  prime  factor  of  each  of  these  numbers. 
Hence,  3x7  will  divide  each  of  them.  As  they  have  no 
other  common  prime  factors,  21  is  their  g.  c.  d. 

Name  all  of  the  common  divisors  of  these  numbers. 
Which  is  the  greatest?     Of  what  is  it  the  product? 

EXAMPLES. 

Find  the  g.  c.  d.  of  the  following: 

1.  24,  28,  36.  8.  210,  294,  462. 

2.  60,  84,  96.  9.  195,  273,  429,  507. 

3.  45,  60,  7.5.  10.  204,  255,  357,  459. 

4.  48,  64,  96.  11.  342,  399,  513,  627. 

5.  28,  42,  56,  98.  12.  295,  413,  531. 

6.  39,  65,  91.  13.  414,  690,  966,  1242. 

7.  112,  110.  108.  14.  780,  234,  312,  390. 

RULE  I. 
For  finding  the  greatest  common  iUriaor. 
Separate  the  ntimhers  into  their  jtritne  factors,  and  find 
the  product  of  those  that  are  common. 


APPENDIX.  381 

The  factoring  method  may  be  employed  satisfactorily  with 
any  numbers,  but  the  process  may  be  shortened  when  the 
numbers  are  large,  by  devices  that  render  some  of  the 
factoring  unnecessary. 

374.     FINDING   THE    G.   C.  D.  BY  AN  EXAMINATION 
OF    DIFFERENCE. 

Illustrative  Example.  Find  the  g.  c.  d.  of  2002,  2366, 
3367. 

A  divisor  of  two  numbers  is  also  a  divisor  of  their  differ- 
ence ;  hence,  the  g.  c.  d.  of  these  numbers  must  also  divide 
the  difference  between  2002  and 
2366.     This  difference  is  364.    Its      ^^^*'* 
prime  factors  are  2,  2,  7,  and  13. 
The  g.  c.  d.  of  2366  and  2002  is      f^  ^ 
also  the  g.  c.  d.  of  364  and  2002.         364  _  2  X  2  x  7  X  13. 
(Why?)     Hence,  we  need  to  compare  only  these  numbers. 
The  prime  factors  of  364  are  2,  2,  7,   13.      By  examining 
2002,  I  find  that  only  three  of  the  prime  factors  of  364  will 
divide   2002 ;   viz.,  2,  7,   13.     The  product  of  these  three 
factors  is  consequently  the  g.  c.  d.  of  364  and  2002,  and 
hence,  of  2002  and  2366  also. 

If  these  factors  are  also  found  in  3367,  their  product  is 
the  g.  c.  d.  of  the  three  numbers.  By  trial  I  find  that  2  is 
not  a  factor  of  3367;  7  and  13  are  prime  factors  of  3367; 
hence,  91  is  the  g.  c.  d.  of  the  three  numbers. 

EXAMPLES. 

(Solve  by  the  above  method.) 

1.  59449,  and  61659. 

AxALTSis.     The  difference  is  2210.     Its  prime  factors  are  2,  5,  13,  17. 
Onlv  13  and  17  are  factors  of  59449 ;  hence  13  X  17  is  the  g.  c.  d. 
Why  need  we  pay  no  attention  to  G1659? 

2.  83971  and  79463. 


381 


NEW  ADVANCED  ARITHMETICc 


The  difference  is  4508.  Its  prime  factors  are  2,  2,  7,  7, 
23.  None  of  these  are  factors  of  79463,  heuce  the  g.  c.  d. 
is  1. 

3.    387  and  2754. 

Multiply  387  by  7.  Is  the  g.  c.  d.  sought  a  divisor  of  this 
product?  Why?  2754  —  27U9  r:^  45.  Vill  the  g.  c.  d. 
divide  45?  Wh}'?  What  are  the  prime  factors  of  45? 
Which  of  these  are  prime  factors  of  387  ?  What,  then,  is 
the  g.  c.  d.  of  387  and  2754?     How  do  you  know? 

*  375.       RULE    II. 

1.  Find  the  difference  between  tico  of  the  mnnbers.  Find 
its  prime  factors.  Determine  irhich  of  them  are  prime 
factors  of  the  smaller  of  the  tiro  }iit)nhers.  Their  product 
is  the  g.  c.  d.  of  the  tivo  numbers. 

2.  Compare  this  product  irifh  a  third  number,  proceed- 
ing as  before,  ami  so  continue  until  all  of  the  numbers 
have  been  disposed  of. 


376.     FINDING    G.  C.  D.   BY   DIVISION. 

Illustrative  Example.     Find  the  g.  c.  d.  of  91  and  325. 

FORM. 

91  )  325  (  3 
273 

~52)91  (1 
52 

39)52(1 
3£ 

"l3  )  39  (  3 
39 

Explanation.  The  g.  c.  d.  of  these  numbers  cannot  be 
greater  than  91.  If  91  will  divide  325,  it  is  the  g.  c.  d.  of 
91  and  325.  The  quotient  is  3,  and  the  remainder  52  ;  hence, 
91  is  not  their  g.  e.  d.     Since  a  divisor  of  a  number  is  a 


APPENDIX.  383 

dhnsor  of  any  of  its  multiples,  the  g.  c.  d.  of  these  numbers 
must  be  a  divisor  of  273.  Since  a  divisor  of  two  numbers 
is  a  divisor  of  their  difference,  the  g.  c.  d.  must  divide  52 ; 
hence,  it  cannot  be  greater  than  52.  Since  52  is  a  divisor 
of  itself,  if  it  will  divide  91,  it  will  divide  273,  by  Principle  1, 
and  325,  by  Principle  2.  The  quotient  is  1,  and  the  remain- 
der 39  ;  hence,  52  is  not  the  g.  c.  d.  sought.  Since  the  g.  c.  d. 
of  91  and  325  must  divide  52  and  91,  it  must  divide  39, 
by  Principle  3  ;  hence,  it  cannot  exceed  39,  If  39  will  divide 
52,  it  will  divide  91,  by  Principle  2;  273,  by  Principle  1; 
and  325,  by  Principle  2.  The  quotient  is  1,  and  the  remain- 
der 13 ;  hence,  39  is  not  the  g.  c.  d.  sought.  Since  the 
g.  c.  d.  must  divide  39  and  52,  it  must  divide  13,  by  Prin- 
ciple 3.  Since  13  will  divide  itself  and  39,  it  will  divide  52, 
by  Principle  2;  91,  by  Principle  2;  273,  by  Principle  1  ;  and 
325,  by  Principle  2  ;  hence,  13  is  the  g.  c.  d.  of  91  and  325. 

377.     RULE   III. 

Select  two  of  the  niiuihers  mid  tiivide  the  greater  by  the 
less,  and  the  less  by  the  renittinder,  if  there  is  one.  Con- 
tinue the  process  until  there  is  no  retnainder.  The  last 
divisor  will  be  the  g.  c.  d.  sought. 

Compare  tJiis  divisor  unth  a  third  nuinber.  proceeding 
in  the  same  manner,  and  thus  continue  until  all  of  the 
numbers  are  disposed  of. 

Note.  Observe  that  this  method  discovers  numbers  tliat  are  smaller 
than  tlie  given  numbers,  and  yet  that  have  the  same  g.  c.  d. 

EXAMPLES. 
Find  the  g.  c.  d.  of  the  following: 

1.  340  and  578. 

2.  333  and  703. 

3.  533,  697,  and  779. 

4.  1265,  1870,  and  8613. 
5-  7944,  12247,  and  13902< 


384  NEW  ADVANCED  ARITHMETIC. 

378.     APPLICATIONS    OP    G.  C.  D. 

1.  What  is  the  greatest  width  that  a  carpet  can  be  to 
prevent  waste  in  covering  the  floors  of  four  rooms  that  are, 
respectively,  15,  18,  21,  and  24  feet  in  width? 

2.  What  is  the  greatest  length  of  floornig  that  can  be 
used,  without  cutting,  for  three  halls  that  are,  respectively, 
24,  36,  and  60  feet  in  length? 

3.  What  is  the  length  of  the  longest  paving-stones  that, 
without  cutting,  may  be  used  to  build  4  walks,  144  feet, 
180  feet,  204  5"eet,  and  300  feet  long? 

4.  What  is  the  capacit}^  of  the  largest  box  that  will  be 
filled  an  integral  number  of  times  in  measuring  160  bushels 
of  oats,  304  bushels  of  wheat,  and  400  bushels  of  rye? 

379.     LEAST    COMMON   MULTIPLE. 
The  Method  for  Large  Numbers. 

In  finding  the  1.  c.  m.  of  large  numbers  that  are  not 
easily  factored,  the  w^ork  may  be  simplified  by  employing 
the  g.   c.  d. 

Observe  that  the  I.  c.  m.  of  two  or  more  numbers  is  the 
product  of  their  g.  c.  d.  and  their  uncommon  prime  factors. 

Illustrative  Example.  Find  the  1.  c  m.  of  6837,  73o3, 
7860. 

(a)  Find  the  g.  c.  d.  of  6837  and  7353  by  the  third 
method. 

(6)    Divide  7353  by  it. 

(c)  Multiply  6837  by  the  quotient. 

(d)  Proceed  in  a  similar  manner  with  the  third  number 
and  the  result  thus  obtained. 

Employ  the  method  with  smaller  numbers  until  the  process 
is  familiar. 

Form  a  rule  from  the  solution  of  the  illustrative  problem. 


APPENDIX.  385 

PROBLEMS. 
Find  the  I.  c.  m.  of  the  following : 

1.  629,  703,  851.  3.    1496,  1768,  2312. 

2.  338,  364,  448.  4.    990,  1305,  1188. 

5.  What  is  the  smallest  sum  of  money  that  may  be  ex- 
pended by  using  only  3-cent  pieces,  5-eent  pieces,  10-cent 
pieces,  or  25-cent  pieces? 

6.  What  is  the  shortest  distance  that  will  exactly  contain 
an  8-foot  measure,  a  12-foot  measure,  a  15-foot  measure,  or 
an  18-foot  measure? 

7.  What  is  the  smallest  quantity  of  oats  that  will  fill,  an 
integral  number  of  times,  a  5-bushel  box,  a  9-bushel  box,  a 
15-bushel  box,  or  a  21-bushel  box? 

8.  What  is  the  product  of  the  1.  c.  m.  of  12,  15,  18,  and 
24,  and  their  g.  c.  d.  ? 

9.  Divide  the  1.  c.  m.  of  7,  12,  21,  9,  10,  and  252,  by  the 
g.  c.  d.  of  80,  120,  840,  and  960. 

10.  What  is  the  difference  between  the  1.  c.  m.  of  10,  45, 
75,  and  90,  and  the  1.  c.  m.  of  7,  15,  25,  and  35? 

11.  What  is  the  shortest  cord  that  could  be  cut  into  pieces 
of  9,  12,  15,  18,  or  45  inches? 

330.     Stone  and  Brick  Work. 

1.  Stone  Work  is  usually  measured  by  the  Perch,  although 
in  many  localities  it  is  estimated  by  the  cubic  foot. 

2.  A  Perch  of  Stone  contains  24|  cubic  feet.  It  is  1  rod 
long,  li  feet  wide,  and  1  foot  thick. 

3.  In  estimating  the  labor  of  laying  stone  and  brick  the 
corners  are  usually  counted  twice,  because  of  the  extra  care 
needed, 

4.  For  8-inch  walls  it  is  customary  to  count  15  bricks  to  the 
foot;  for  12-inch  walls,  21  bricks  are  counted  for  a  foot. 


386  NKW  ADVANCED  AIUTllMirriC. 

381.     Grain. 

A  cubic  foot  is  about  .«  (jf  m  bushel.  Tlic  capacity  of  a 
waj(oii-box  or  a  bin  may  bo  found  approxiiiiati'ly  l)y  liii(liu<^ 
the  number  of  cubic  feet  wliich  it  contains,  and  multiplying 
tliis  result  by  .«. 

382.     Ear  Corn. 

A  iMishel  of  ear  corn  (••Milains  aliont  2]  eubic;  feet.  To 
rnid  I  lie  iinnil)cr  (if  bushels  a  bin  will  contain,  lind  the  num- 
bci  (.f  enl)ic  leet  in  the  bin  and  lake  I  of  it. 

383.  Hay. 

Hay  measurement  is  only  :ippr<K\imately  correct.  About 
'^:^\  cubic  fi-ct  of  well-settled  timothy  hay  will  weigh  a  ton. 
If  only  partially  settled,  at  least  4.jO  cubic  feet  must  be 
allowid.  Clover  liny  is  much  lighter.  Allow  about  ooO 
cubic   feet  t<j  the   Ion. 

384.  Coal. 

Lchiirh  stove-coal  and  Schuylkill  white-ash  stove-coal  con- 
tain iiboiil  a.")  cubic  feet  to  the  ton.  Other  varieties  vary 
but  slightly  from  this  estimate^. 

385.     MARINERS'    MEASURE. 

6    feet  =  1  fathom. 

120    fathoms  =  1  cal'h;  length. 

7',  cable  lengths  =  1  mile. 

386.     AVERAGE    OF    ACCOUNTS. 

A  problem  in  Average  of  Accounts  is  a  double  problem  in 
K<|uation  of  Payments. 

The  two  sides  of  the  account  may  be  compared  by  assum- 
ing a  <'oinmon  day  of  settlement. 


APPENDIX.  387 

Illustrative  Problem. 
Dr.     A.  B.  Smith  in  acct.  with  Brown  Bros.     Cr. 


1891. 

Jan.     1.  To  mdse.,  $368.25 

28.  "       "         225.00 

Mar.  12.  "       "         18G.50 


1891. 

Feb.  15.  By  cash,  8450.00 
Apr.  12.  "  mdse.,  400. 0(> 
May  24.     "    produce,  158.00 


Analysis.  Let  us  assume  that  A.  B.  S.  paid  his  account  in  full  on 
May  24.  He  would  then  have  had  a  credit  of  4  mo.  23  d.  on  the  first,  3  mo. 
26  d.  on  the  second,  and  2  mo.  12  d.  on  the  third.  If  interest  were  paid  at 
6%,  the  charge  on  the  first  would  he  S8.78;  on  the  second,  $4.35;  on  the 
third,  S2. 24.  Their  sum  is  S15. 37.  'J'he  sum  of  the  items  is  S779.75.  If 
interest  were  charged,  then,  on  May  24,  A.  B.  S.  would  pay  to  Brown 
Bros.  $779.75,  and  Si 5.37  as  interest. 

If  Brown  Bros,  were  to  pay  A.  B.  S.  on  May  24,  they  would  have  a 
credit  of  3  mo.  9d.  on  the  first,  and  1  mo.  12  d.  on  the  .second.  If  interest 
•were  paid  at  6%,  the  charge  on  the  first  would  he  S7.43 ;  on  the  second, 
$2.80.  Their  sum  is  S10.23.  The  sum  of  the  items  is  51,008.  If  Brown 
Bros,  settled  with  A.  B.  S.,  then,  on  May  24,  they  would  pay  him  51,008, 
and  SI 0.23  as  interest.  But  Brown  Bros,  owe  A.  B.  S.  S228.25  more  than 
he  owes  them.  If  there  were  no  interest  charge,  they  could  settle  by  pay- 
ing him  that  balance.  Since  interest  is  to  be  considered,  they  would  lose 
S15.37,  and  A.  B.  S.  would  lose  810.23.  Their  loss  would  e.xceed  his  by 
S5.14.  To  prevent  this  lo.ss,  they  should  retain  the  S228  25  until  it  would 
earn  them  55.14  at  G%.  The  intere.«t  on  5228.25  is  about  5  038  per  day. 
It  will  take  130  days  for  5228.25  to  gain  55.14 ;  hence.  Brown  Bros,  should 
retain  the  balance  until  Oct.  7. 


387.     ORIGIN  OF  UNITS. 

1.  The  yard  is  the  standard  from  'vvhich  nearly  all  of  our 
units  of  measure  are  derived.  It  was  definitely  fixed  by 
what  is  known  as  the  "  Pendulum  experiment." 

2.  A  pendulum  was  found  which,  at  London,  at  the  sea 
level,  vibrated  once  in  a  second.  It  was  divided  into  391 ,393 
equal  parts,  of  which  360,000  were  about  equal  to  the  yard 
then  most  commonly  used.  By  law  this  became  the  unit  of 
linear  measure,  under  the  name  of  the  vard.      It  was  divided 


/ 


388  NEW  ADVANCED  ARITHMETIC. 

into  three  equal  parts  called  feet,  and  each  of  these  was 
divided  into  twelve  equal  parts  called  inches. 

3.  Since  a  square  foot  is  a  square,  each  of  whose  sides  is 
a  linear  foot,  it  is  seen  that  the  units  of  square  measure  are 
derived  from  the  same  experiment. 

4.  Since  a  cubic  foot  is  a  cube  each  of  whose  edges  is  a 
linear  foot,  the  units  of  cubic  measure  have  the  same  origin. 

5.  A  gallon  is  231  cubic  inches.  The  bushel  is  similarly 
derived.  The  standard  of  weight  is  the  Troy  pound.  Its 
weight  is  equal  to  the  weight  of  about  22.8  cubic  inches  of 
distilled  water. 


388.     THE    METRIC    SYSTEM    OF  WEIGHTS  AND 

MEASURES. 

1.  On  account  of  the  great  variety  of  scales  employed 
in  the  common  system  of  weights  and  measures  in  use  in 
this  country,  an  effort  has  been  made  to  introduce  the  French 
system. 

2.  The  standard  unit  is  the  Meter,  which  was  supposed  to 
equal  one  ten-millionth  of  a  quadrant  of  the  meridian  passing 
through  Paris.  It  is  about  |  of  a  yard ;  but  those  using  this 
system  must  learn  to  think  in  its  units. 

3.  Decimal  parts  of  the  meter  are  indicated  by  prefixing 
to  meter,  milli,  meaning  one-thousandth,  centi,  meaning  one- 
hundredth,  and  deci,  meaning  one-tenth. 

4.  Decimal  multiples  of  the  meter  are  expressed  by  pre- 
fixing to  meter,  deka,  meaning  ten,  hecto,  meaning  one 
hundred,  kilo,  meaning  one  thousand,  and  myria,  meaning 
ien  thousand. 

Note. — The  first  three  prefixes  are  Latin,  and  the  otliers  arc  Greek 

The  abbreviations  for  names  containing  the  Latin  prefixes 
are  printed  in  small  letters,  and  those  containing  the  Greek, 
in  capitals. 


APPENDIX.  3b9 

389.     Table  of  Long  Measurt. 

10  millimeters  (ram.)  equal  1  cenlimeler      (fin.)  equal  .3937+  in. 

10  centimeters  ••  1  decimeter      (dm.)  ••  .■5.!)37+  in. 

10  decimeters  "  1  imier              (m.)  "  3!j.37+  in. 

10  meters  "  1  dekametcr     (Dm.)  "  32.8+ ft. 

10  dekanieters  •'  1  hektomelcr  (llin.)  •'  19.927+ rd. 

10  hektomelers  "  I  kilometer         (l\n\.)  "  .()21+ mi. 

10  kilometers  "  1  myriameter  (Mm.)  "  G.213+  mi. 

Note. — The  uuits  most  commonly  used  are  printed  in  italic. 

1,  Long  distances  are  measured  in  kilometers.  This  unit 
is  about  g  of  a  mile. 

2.  The  symbol  denoting  the  denomination  may  be  placed 
after  the  integral  part  of  the  expression ;  thus  24  m.  oG,  or 
after  the  entire  expression,  24. 5G  m. 


390.     Surface  Measure. 

The  surface  units  are  squares  each  of  whose  sides  is  a  linear 
unit.  It  follows  that  100  of  each  order  make  one  of  the 
next  higher. 

TABLE. 

100  sf|.  mm.  equal  1  sq.  cm. 

100  sq.  cm.        ''      1  sq.  dm. 

100  sq.  dm.       "      1  sq.  m.      =1.190  sq.  yd. 

100  sq.  m.         "     1  sq.  Dm.  —  119. P.  +  s(|.  yd. 

100  sq.  Dm.      "     1  sq.  Hm.  =  2.47  +  A. 

100  sq.  Hm.      "     1  sq.  Km.  =  247.114  A. 

1.  The  square  meter  is  used  in  the  measurement  of  small 
surfaces  as  the  square  yard  is  used  in  the  ordinary  system. 
When  used  to  measure  land  it  is  called  a  centare  (en). 

2.  The  sq.  Dm.  is  called  an  are  when  used  as  a  land 
measure,  and  the  sq.  Hm.  a  hectare  when  so  used. 

26A 


390  NEW  ADVANCED  ARITHMETIC. 

391.     Measures  of  Volume. 

The  voliune  units  are  cubes,  each  of  whose  edges  is  a 
linear  unit.  In  the  following  table  1000  of  each  deuomi- 
nation  make  one  of  the  next  higher. 

TABLE. 
1000  cu.  mm.  equal  1  cu.  cm. 
lUUO  cu.  cm.  "  1  cu.  dm. 
1000  cu.  dm.      "     1  cu.  m.  =  35.31  G  cu.  ft. 

1-    The  cu.  m.  is  the  unit  most  commonly  used. 

2.  "When  the  cubic  meter  is  used  in  measuring  wood  it  is 
called  a  stere.  One  tenth  of  a  stere  is  a  decistere.  10  steres 
make  a  deJcaster  (Dst.) 

3.  The  stere  is  a  little  more  than  a  quarter  of  a  cord. 

392.     Tables  of  Liquid  and  Dry  Measure. 

1  milliliter  (ml.)  =  1  cu.  cm. 

10  ml.  —  1  centiliter  (cl.) 

10  cl.  =  1  deciliter  (dl.) 

10  dl.  =  1  liter  (1.)  =--  1  cu.  dm. 

10  1.  1=  1  decaliter  (Dl.). 

10  Dl.  =  1  hectoliter  (HI.) 

10  HI.  =  1  kiloliter  (Kl.)  =  1  cu.  m. 

10  Kl.  =  1  myrialiter. 

Dry.  Liquid. 

1  liter  (1.)  =     .90H  qt.  =  l.()')7  qt. 

1  decaliter  (Dl.)  =    1.13;')  pk.  =r  2,642  gal. 

1  hectoliter  (HI.)  =  2.837  bu.  =  26.417  gal. 

1  kiloliter  (Kl.)  =  28.37  bu. 

The  liter,  which  is  very  nearly  the  liquid  quait,  and  the 
hectoliter  —  about  2g  bushels  —  are  the  units  most  commoulv 
used 


APPENDIX.  391 

393.     Weights. 

The  unit  of  weight  is  the  (jram^  which  is  the  weight  of  one 
cu.  cm.  of  pure  water  at  the  temperature  of  greatest  density. 

TABLE. 
10  milligrams  (mg.)  =  1  centigram  (eg.) 
10  eg.  —  1  decigram  (dg.). 
10  dg.   =  1  gram  (g.) 
10  g.      =1  decagram  (Dg.). 
10  Dg.  =  1  iiectogram  THg,) 

10  Ilg.  —  1  kilogram  (Kg.)  —  wt,  1  cu,  dm.  of  water. 
10  K.     =  1  myriagram  (Mg.). 
10  Mg.  =  1  quintal  (Q.) 
10  Q.     =1  tonueau  (T.)  =■  wt,  1  cu.  m.  of  water. 

1  gram  (g.).  =  15.432  +  grains 

1  kilogram  (Kg.)   =  2.204  +  Ih.  a\% 
1  tomieau  (T.)       =  2204.621  +  lb.  av. 

1.  The  units  most  commonly  used  are  the  the  centigram, 
the  gram,  the  kilogram,  and  the  tonueau. 

2.  The  kilogram  is  called  the  kilo,  for  brevity. 

394.     Table  of  Approximate  Equivalents. 

1  decimeter  =  4  in.        nearly. 

1  meter  =  1^  yd.         " 

1  decameter  =  2  rd.  " 

1  kilometer  =  f  of  1  mi.   " 

1  are'  =  Jg^of  an  A.  " 

1  stere  =  ^  of  a  cord." 

1  liter     '  =  1  liq.  qt.     " 

1  hectoliter  =  3  bushels    " 

1  gram  =  15i  grains  " 

1  kilo  =  2^ 'lb.  av.   "  > 

1  tonueau  =  W^  tons.    " 


392  NEW  ADVANCED  ARITHMETIC. 

395.    Exercises  on  the  Metric  Tables. 

1.  Reduce  2864  m.  to  mm. ;  to  dm. ;  to  Hm. ;  to  Mm. 

2.  Reduce  24685  sq.  dm.  to  sq.  cm. ;  to  sq.  Dm. ;  to 
sq.  Km. 

3.  Reduce  2.5709853  cu.  dm.  to  cu.  cm. ;  to  cu.  m. ;  to 
decisteres;  to  dekasters. 

4.  Reduce  47073  1.  to  c^  ;  to  Dl. ;  to  Kl.  ;  to  dl. 

5.  Reduce  279436  Dg.  to  g. ;  to  mg. ;  to  Mg. ;  to  Q. 

6.  How  many  sq.  m.  in  the  floor  of  a  room  12  m.  long  and 
10  m.  wide? 

7.  Find  the  length  of  j-our  school-room  in  meters.  Find 
the  width  of  your  desk  in  decimeters.  What  is  the  area  of 
the  top  of  your  desk  in  square  decimeters?  It  is  what  part 
of  a  sq.  m.  ? 

8.  Find  the  number  of  square  centimeters  in  a  pane  of 
glass  in  a  window  of  your  school-room.  What  part  of  a 
sq.  m.  is  it? 

9.  Lay  off  in  your  school-yard  a  square  a  dekameter  on 
a  side.  What  is  this  unit  used  for?  How  many  square 
meters  does  it  contain?  What  is  its  name?  Calling  its 
area  120  square  yards,  what  part  of  an  acre  is  it? 

10.  Find  the  volume  of  your  school-room  in  cubic  meters. 
If  it  were  filled  with  wood  how  many  steres  would  it  contam? 
About  how  many  cords  ? 

11.  What  would  it  cost  to  lath  and  plaster  the  ceiling  of 
your  school-room  at  28  cents  a  sq.  m.  ? 

12.  If  your  school-room  were  used  for  a  grain  bin,  how 
many  hectolitei-s  of  shelled  corn  could  be  put  into  it?  How 
many  bushels  ?  If  it  were  a  tank  how  many  decaliters  of 
water  wduld  it  hold?     How  many  gallons? 

13.  How  many  grains  are  there  in  a  gram?  How  many 
grams  in  an  ounce  avoirdupois  ?  A  gram  is  about  what  part 
of  an  ounce?    Of  a  pound? 


APPENDIX.  393 

14.  Compare  the  kilo  with  the  pound ;  the  touiieau  with 
the  ton. 

15.  Find  the  cost  of  10  kilos  of  coffee  at  3  cents  an  Hg. 

16.  A  field  is  1  kilometer  in  length  and  5  hectometers  la 
width.     What  is  it  worth  at  82.25  an  are? 

17.  What  is  the  weight  of  a  liter  of  i)ure  water? 

18.  The  atmospheric  pressure  under  ordinary  conditions 
is  about  1.5  pounds  to  the  square  inch.  What  is  it  in 
tonneaus  per  square  meter? 

19.  How  many  bushels  in  a  quintal  of  wheat?  (Wheat 
60  lbs.  to  bu.) 

20.  What  will  a  liter  of  mercury  (specific  gravity,  130 
weigh?     Find  weight  also  in  pounds. 

21.  What  is  the  cost  of  a  pile  of  wood  12  m.  long,  2  m. 
v/ide,  and  2  m.  high,  at  Si. 25  a  stere? 

22.  A  bin  is  10  m.  long,  3  m.  wide,  and  3  m.  high.  It  is 
filled  with  wheat.  What  is  the  value  of  the  vrheat  at  210 
cents  an  HI.  ? 

23.  Find  the  weight  in  quintals  of  the  above  wheat,  count- 
ing it  I  as  heavy  as  water. 

24.  Find  the  capacity  in  HI.  of  a  cylindrical  cistern  whose 
diameter  is  3.2  m.,  and  depth  4.1  m. 

Note. — The  hectoliter  is  about  |  of  a  barreL 


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OCT  2  1934 

JAN  24   1938 


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